Why do Jehovah's Witnesses make the Old Testament forms of address to God ("Jehovah" and "Father") more important than "Abba?"

The reason why Jehovah's Witnesses relegate the form of address, "Abba" to almost non-use, is due to their doctrine that only 144,000 humans have ever had the right to call God their "Abba". As (on average) 99% of all Jehovah's Witnesses for the past 100 years deny being part of that group of 144,000, they cannot (dare not) address God as their "Abba". That is only the entitlement of the tiny number they say are anointed of 'holy spirit' so as to have 'the heavenly calling'.

It might be that someone attending their annual solemn memorial of Christ's death could hear "Abba" during a prayer - if the person praying was a partaker of the bread and wine at that event. Anointed, in other words - expecting to go straight to heaven when they die. But he could only mean himself, and not the others. (And it would be a 'he' praying, not a 'she', because women are not allowed to pray at their public meetings when men are present.)

The basis for this is Romans chapter 8, where all who belong to Jesus Christ are assured that they have the indwelling Holy Spirit as their guarantee that they belong to Christ and will be with him in glory. Verse 9 addresses those who have the Spirit of God living in them. Verse 10 warns that if anyone does not have the Spirit of Christ, they do not belong to Christ. The following verses go on to assure Christians who have the indwelling Holy Spirit that they will live as "sons of God"; they received "the spirit of [adoption] as sons" and can cry out "Abba, Father!". The Spirit testifies with their spirit "that we are God's children" therefore joint heirs with Christ". (N.W.T.)

But those verses in Romans chapter 8 only apply to those who take the bread and wine at their "Memorial" service, once a year. That leaves many millions of Jehovah's Witnesses unable to call God their "Abba". So, they don't. And it is true. They cannot call God their "Abba", because those ones are not indwelt with the Holy Spirit, anointed, called to be with Christ in heaven. Only those in the 144,000 class are such spirit-filled sons of God, say all Jehovah's Witnesses.

There's no need to explore what they say about anything else. It's not about a "requirement" to call God 'Abba' - it's about the "entitlement" to call God 'Abba'. 99% of them do not believe they are entitled to call God 'Abba', and that is actually true. They are not entitled, otherwise they would take the bread and wine at their annual Memorial.

References: Wt.B&T.Soc., Insight on the Scriptures Vol.1 p13 under Abba - "Romans 8:15... the word is used in connection with Christians called to be spirit-begotten sons of God and indicates the intimacy of their relationship with their Father... they are also sons in the house of a loving Father, and they are made positively aware of this status by the holy spirit through their Lord Jesus."

Also p.270, 'Partakers & Other Attenders not partaking.'

Also p.1197 col.2 lines 6-17, these anointed ones will reign with Christ in heaven and are born again. That only applies to the 144,000, as all JWs know.

Two other occurrences of the use of the word [Abba] are in the apostle Paul’s letters, at Romans 8:15 and Galatians 4:6. In both places the word is used in connection with Christians called to be spirit-begotten sons of God and indicates the intimacy of their relationship with their Father. While they are “slaves to God,” yet they are also sons in the house of a loving Father, and they are made positively aware of this status by holy spirit through their Lord Jesus. https://wol.jw.org/en/wol/d/r1/lp-e/1966567

NOTE: Jehovah's Witnesses believe that only the remnant of the 144,000 are "spirit-begotten sons of God" who alone have a heavenly calling.

Jehovah's Witnesses do not ascribe different importance between the so-called Old and New Testaments. They view the Bible as one unified message. Certainly, the Law of the Hebrew Scriptures was brought to a finish by Jesus Christ, but there are also many timeless facts contained in the Hebrew scriptures about God, one of which is his name.

I am Jehovah. That is my name. — Isaiah 42:8

Then God said once more to Moses: “This is what you are to say to the Israelites, ‘Jehovah the God of your forefathers, the God of Abraham, the God of Isaac, and the God of Jacob, has sent me to you.’ This is my name forever, and this is how I am to be remembered from generation to generation.
— Exodus 3:15

May people know that you, whose name is Jehovah, You alone are the Most High over all the earth.
— Psalm 83:18

On the other hand, the Bible does not teach that "Father" is a personal name of God, but rather it is used as a title, just as your own fleshly father's personal name is not Father, though the word functions grammatically as a proper noun when you're addressing him with that title.

Christians can and should address God as their Father, as exemplified by Jesus, but nowhere does any scripture say that we should stop using God's personal name, Jehovah. The Bible teaches that God's name is the same for all eternity.

For “everyone who calls on the name of Jehovah will be saved.” — Romans 10:13