Luke's introduction, doesn't seem like he is claiming he was inspired?

When reading the introduction of the Gospel of Luke, it seems very clear that he isn't really claiming to be inspired by God

In Luke 1:1, the author says that he is writing down "those things which are most surely believed among us." At most, this is an affirmation that the Lukan community believes those things, with no claim that he was writing down those things known certainly to be true, and therefore no claim to inspiration.

So why were those things believed "among us" in the Lukan community? Luke 1:1 extract: "They delivered them unto us." These things came through a chain of sources, the earliest of whom the evangelist believes to have been eyewitnesses (Luke 1:2 extract): "from the beginning were eyewitnesses, and ministers of the word." He does not say he ever met these eyewitnesses, in fact to have done so would make "from the beginning" incongruous. Once again, the evangelist says he is writing down what others assure him to be true, not what God inspired him to write.

In Luke 1:3, the evangelist says that he had perfect understanding of the things he had been told, which is why he is writing to Theophilus. This suggests that the evangelist is more worthy of writing these accounts, perhaps more than some others in the community, but does not mean he claims to be inspired.

2 Timothy 3:16 is frequently cited as evidence that all scripture is inspired by God, which then includes Luke's Gospel.

2 Timothy 3:16 (NAB): All scripture is inspired by God and is useful for teaching, for refutation, for correction, and for training in righteousness,

Three problems arise in this understanding of 2 Timothy:

1. The great majority of New Testament scholars believe that Second Timothy was not written by Paul or in Paul's lifetime, which means that verse 3:16 is not an inspired guide.
2. If Second Timothy really was written by Paul, then he wrote it before Luke's Gospel was written and therefore did not have that Gospel in mind.
3. The New Testament canon was not determined in the first century, so 2 Timothy 3:16 must only have been referring to the Old Testament scriptures.

The Catholic Church, unlike the Protestant Churches, regards its teachings and traditions as equal in importance to the Bible. It might be believed that Luke was an inspired writing, but Catholics will arguably be less affected than other Christians if this was not the case, as long as the Church's own traditions are sacred.

I do not know the Catholic position, but an idea I have seen expressed by many sources is that Luke was a traveling companion of Paul, an Apostle. Thus Luke's work was supervised and approved by an apostle, granting it apostolic authority. Apostolic authority is something considered important by the Catholic church.

Some citations:

https://goexplorethebible.com/blog/adults/how-did-we-get-the-new-testament-canon/

https://www.christianity.com/jesus/life-of-jesus/harmony-of-the-gospels/1-the-prologue.html

The Apostle Paul confirms to us that the Gospel according to St. Luke is Scripture:-

"For the scripture saith, thou shalt not muzzle the ox that treadeth out the corn. And, The labourer is worthy of his reward." (1 Timothy 5:18)

The second scripture is a quote from Luke's Gospel:-

"And in the same house remain, eating and drinking such things as they give: for the labourer is worthy of his hire. Go not from house to house." (Luke 10:7)

You might have thought it to be a quote from Matthew's Gospel:-

"Nor scrip for your journey, neither two coats, neither shoes, nor yet staves: for the workman is worthy of his meat." (Matthew 10:10)

But a comparison of the Greek shows Paul's text is quoting Luke where "hire" or "wages" is used rather than Matthew which uses "meat" or "nourishment".

There are two possible reasons why the Holy Spirit ensured Paul tells us Luke's Gospel is Scripture: it is generally believed Luke was not a Jew and "salvation is of the Jews" (John 4:22); and, of course, Luke was not an Apostle, and so his Gospel needed Apostolic confirmation.

Here are copies from online Greek interinear:

Relevant part of 1 Timothy 5:18 :- kai axios ho ergatEs tou misthou autou - and WORTHY THE ACTer/worker OF-THE hire/wages OF-him

Part of Matthew 10:10 - axios gar ho ergatEs tEs trophEs autou estin - WORTHY for the ACTer/worker OF-THE nourishment OF-him is

Part of Luke 10:7 - axios gar ho ergatEs tou misthou autou estin - WORTHY for the ACTer/worker OF-THE hire/wages OF-him is

(All interlinear quotes from https://www.scripture4all.org/OnlineInterlinear/Greek_Index.htm)

Incidently, this also shows us that Luke's Gospel was written before Paul's 1st letter to Timothy.

I think you are correct, Luke did not write his books believing that they were on the same level as the Holy Scriptures he was taught as a child. But I don't think this makes the Bible (or his two books) any less "inspired" by God. I imagine that many authors did not write their books with the intent that they become part of some "holy compilation." Psalms is a compilation of David's (and a few others) song lyrics. Proverbs could have been written by Solomon just to share what he believed to be sound advice. To me, and I think most Christians, calling the Bible the word of God is merely recognizing the authority of its authorship: leaders appointed by God, or men chosen by God to speak on His behalf, who felt compelled to write down their experiences that were very much influenced by God.

But I think no discussion of what the Bible is and how it ought to be used and perceived is complete without mentioning what Paul said about the Scriptures:

2 Timothy 3:14-17
14 But as for you, continue in what you have learned and have become convinced of, because you know those from whom you learned it, 15 and how from infancy you have known the Holy Scriptures, which are able to make you wise for salvation through faith in Christ Jesus.
16 All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness, 17 so that the servant of God may be thoroughly equipped for every good work.

So according to Paul, an authority on Christianity, the Bible is a foundation upon which to teach and train men in "righteousness". He also says that it is "God-breathed," which is important because the word "inspire" derives from the phrase "to breathe life into". The words within the Bible are those of men, but it is God who "breathes" life into them. So when we say that the Bible is the inspired word of God it is because the words themselves are useless and meaningless without God to breathe life into them. And I find that He continues to breathe life into old scriptures; as it is written: "Behold, I make all things new!"

Christians do not believe that the Bible was dictated by God in the same sense that Muslims believe the Quran was dictated to Muhammed. They believe that God inspired the writers, through their own knowledge and personalities, to write the things that God wanted his people to know and remember. Because of this it is possible for an author to write inspired scripture without even being aware of it. It is also true that the early church spent hundreds of years in prayer, in research, and in consideration before choosing the books that today make up the Bible, and the belief is that the process was also inspired by God through his Spirit.

The introduction to Luke does not in any way contradict this. Luke (and it's sister book, Acts) is a carefully researched and collated story of the Life of Jesus and the early church. It is also inspired by God, who worked within the life and thoughts of Luke to bring these writing to fruition. This issue is not restricted to Luke. Many of the other books are letters written by Christian leaders to their churches. It is likely that they did not know of God's working through them, or that their writings would one day become holy scripture. However that does not make it any less true.