Why was Jesus not welcomed and carefully tested by the Jews if they were (and are) actually waiting for Messiah?

Your question is not naive. Nor is there a single answer.

1. No sign. Some Jews expected the Messiah to perform an outstanding miracle. Moses prophesied this:

15 The Lord your God will raise up for you a prophet like me from among you, from your fellow Israelites. You must listen to him. (Deuteronomy 18:15)

Since Moses performed ten plagues against Egypt, parted the Red Sea, called down Manna from heaven, received the Ten Commandments, and was spoke to by God in the presence of witnesses, this set their expectations. Jesus performed many healings, but nothing as spectacular as Moses - except for raising Lazarus and Jairus' daughter from the dead and then himself rising from the dead. However - no human witnessed Jesus rise, and only his disciples saw him after he rose and saw him ascend into heaven. The general Jewish population and the leaders never saw that sign.

Indeed, Jesus told them that they would not get the sign that they wanted:

29 As the crowds increased, Jesus said, “This is a wicked generation. It asks for a sign, but none will be given it except the sign of Jonah. 30 For as Jonah was a sign to the Ninevites, so also will the Son of Man be to this generation. 31 The Queen of the South will rise at the judgment with the people of this generation and condemn them, for she came from the ends of the earth to listen to Solomon’s wisdom; and now something greater than Solomon is here. 32 The men of Nineveh will stand up at the judgment with this generation and condemn it, for they repented at the preaching of Jonah; and now something greater than Jonah is here. (Luke 11:29-32)

The thing about the sign of Jonah is that it was a sign to Gentiles, the Ninevites.

2. Political expectations. The Jews expected a political leader who would lead them in a revolt and free them from Roman rule. Some people were willing to entertain the idea that Jesus could be the Messiah, but his response to them turned them off:

14 After the people saw the sign Jesus performed, they began to say, “Surely this is the Prophet who is to come into the world.” 15 Jesus, knowing that they intended to come and make him king by force, withdrew again to a mountain by himself. (John 6:14-15)

Jesus' disciples were not free from this expectation:

6 Then they gathered around him and asked him, “Lord, are you at this time going to restore the kingdom to Israel?” (Acts 1:6)

3. Doctrinal divisions. The Pharisees believed in an afterlife and that God can ressurect people, but the Sadducees did not:

18 Then the Sadducees, who say there is no resurrection, came to him with a question. (Mark 12:18)

4. Source of authority. The Jews questioned Jesus about where he received his authority. His answer forced them to accept him on faith, not a sign.

23 Jesus entered the temple courts, and, while he was teaching, the chief priests and the elders of the people came to him. “By what authority are you doing these things?” they asked. “And who gave you this authority?”

24 Jesus replied, “I will also ask you one question. If you answer me, I will tell you by what authority I am doing these things. 25 John’s baptism—where did it come from? Was it from heaven, or of human origin?”

They discussed it among themselves and said, “If we say, ‘From heaven,’ he will ask, ‘Then why didn’t you believe him?’ 26 But if we say, ‘Of human origin’—we are afraid of the people, for they all hold that John was a prophet.”

27 So they answered Jesus, “We don’t know.”

Then he said, “Neither will I tell you by what authority I am doing these things. (Matthew 21:23-27)

Interestingly enough, John the Baptist, growing impatient and disillusioned while in prison, also questioned Jesus through his emissaries:

18 John’s disciples told him about all these things. Calling two of them, 19 he sent them to the Lord to ask, “Are you the one who is to come, or should we expect someone else?”

20 When the men came to Jesus, they said, “John the Baptist sent us to you to ask, ‘Are you the one who is to come, or should we expect someone else?’”

21 At that very time Jesus cured many who had diseases, sicknesses and evil spirits, and gave sight to many who were blind. 22 So he replied to the messengers, “Go back and report to John what you have seen and heard: The blind receive sight, the lame walk, those who have leprosy are cleansed, the deaf hear, the dead are raised, and the good news is proclaimed to the poor. 23 Blessed is anyone who does not stumble on account of me.” (Luke 7:18-23)

Jesus was asking John if he believed the prophets. All the things listed were associated with the coming of the Messiah. Jesus was calling on John to trust the Word of God as his sign. This is similar to the parable Jesus told about the rich man and Lazarus. The rich man wants a sign to be given to his brothers to save them from Hell, but Jesus says that the Words of Moses and the Prophets should be enough.

31 “He said to him, ‘If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead.’” (Luke 16:31)

5. The Law, misinterpreted. Following from the question about authority, are issues arising from Jesus' and his disciples behavior. Healing on the sabbath, eating grain on the sabbath, associating with tax collectors and sinners, forgiving a sinful woman, and related things. Jesus as the very embodiment of God's law is free to interpret it in a perfect way, guaranteed to diverge from imperfect human traditions. Since there were so many ways that their interpretation differed from God's original intent and the law of mercy, Jesus was bound to do things of which they disaproved.

6. God's Sovereign will. The Scribes, Pharisees, and Saduccees sent people to question Jesus frequently. They did test him with their questions repeatedly. But it was God's sovereign will that Jesus be rejected.

22 “Fellow Israelites, listen to this: Jesus of Nazareth was a man accredited by God to you by miracles, wonders and signs, which God did among you through him, as you yourselves know. 23 This man was handed over to you by God’s deliberate plan and foreknowledge; and you, with the help of wicked men, put him to death by nailing him to the cross. (Acts 2:22-23)

Paul has a lot to say about this in Romans 9, but that is a lot of Bible to quote. Paul says a lot about election and God's purpose in using the gift of salvation to the Gentiles in order to make the Jews jealous and ultimately save them.

So God brought about a cultural situation in Israel with so many conflicting forces that despite a huge number of miracles (tens of thousands healed, fed, freed from possession, etc.) that he would be able to do all that and yet demonstrate the hardness of the people's hearts, just as he did with Pharaoh. Salvation comes to the repentant, and Jesus needed to demonstrate the magnitude of the sin in their hearts in order to show just how bad we all need a savior. The messiah does not come without cost.

7. A new order of Growth. Moses introduced the people to God in this order:

   • Miraculous external acts of power - the Exodus
   • Emotional changes - rejoicing, loyalty, trust in Moses' leadership
   • God's Law - delivered on Sinai

Jesus, however, called for things to proceed in reverse. Small seeds grow into big trees. In Jesus' parable of the soils in Matthew 13, Satan first tries to snatch the word (the Law), then scare people through persecution and hardship (emotion), then confuse our priorities in matters related to how we interact with the world. This suggest the following order of Jesus' work among people:

• The Son of God / the Truth - transforms your mind
• The Holy Spirit / Comforter - reorders your emotions, provides comfort, courage, hope
• God the Father / Provider - reorders your priorities and sovereignly provides your needs, sometimes through acts of power

Jesus led with the truth; following his death he sent the Holy Spirit, and only then began to act in power through his church. The Jews were expecting the Messiah to follow the same order as Moses, starting with the big things, but instead, he started with the small things andmade them grow.

Part of the OP assumption is the Jews in leadership didn't test Christ. They did, but they didn't like what they found.

And no man was able to answer him a word, neither durst any man from that day forth ask him any more questions. Mt 22:46

Not to be callous, but the reason the leaders didn't acknowledge the Messiah was because He was going to flatten their authority, money, and respect.

[After raising Lazarus out from the dead] The Pharisees therefore said among themselves, Perceive ye how ye prevail nothing? behold, the world is gone after him. Jn 12:19

If we let him thus alone, all men will believe on him: and the Romans shall come and take away both our place and nation. Jn 11:48

So, the leaders of the day did test Jesus. They actually found Him to be the Messiah. He answered all of their questions. He healed the sick, bound the lame, forgave sins, and fulfilled the prophecies about Messiah. They eventually realized what that truly meant; loss of their power, prestige, and powerbook. So they had their Messiah crucified and buried.

He was tested

The Jews living in the time of Jesus did try to discern whether Jesus was the real Messiah that they had been yearning for, because they disliked living under Roman rule and wanted to go back at least to the Hasmonean period, or better yet, to the David & Solomon period (the golden age).

They were waiting for God to provide them with what God Himself promised in His covenant with David (2 Sam 7:12-17), confirmed through His prophet Isaiah hundreds of years later ("a shoot from Jesse's stump", Isa 11:1-10). But as the waiting grew to hundreds of years after Isaiah, theories of what form the Messiah would take abounded. One theory even predicted 2 messiahs, one from the line of David, another from the line of Aaron. False pretenders came and went. By the time of Jesus the yearning didn't go away, it only became stronger.

Many passages in the Gospel showed how they approached Jesus, observed what He said and did, saw His miracles, heard testimonies including by John the Baptist, asked for miraculous signs, etc. They were excited for a while, but when Jesus began to claim that He was the Son of God and broke what they thought to be the right way of observing the Sabbath (and many other rules added to the Torah by the Pharisees) they began to have doubts and thought that maybe Jesus was another false pretender.

The lack of a warm welcome

It was because Jesus did not meet their expectations.

The climax was when Jesus was crucified. It was a scandal. A messiah shouldn't die a shameful death. That meant God was not with him. The problem was that they didn't read OT the way Christians do. Many passages in the OT continue to be interpreted differently by the Jews of today (just see Judaism.SE for yourself, such as this answer, to see how different the interpretations can be). For example, the Suffering Servant prophecy in Isa 53, so critical to Christian theology to identify Jesus as the Messiah sent by God, is interpreted to refer to the nation of Israel which had been displaced and persecuted throughout thousands of years (the 20th century Holocaust only made that interpretation stronger). The Jews then as well as today don't believe that God resurrected Jesus from the dead. To them then as well as today, Jesus was either a lunatic or if not a lunatic he committed the most serious blasphemy by claiming to be God. If Jesus was the latter, by that calculation Jesus deserved to die, since God commanded them to treat false prophets that way (Deut 13:1-5).

I hope reading the Gospels against the background above helps answer all your questions. Read especially how Jesus responded to them.
Basically, they were too close minded. They didn't want to believe Jesus's "more than enough" signs and wonders and testimonies, and Jesus & the Gospel writers chalked it up to "hardness of heart", predicted by Isaiah as well (Matt 13:12-17 quoting Isa 6:9-10).