I addressed Tit 3:3-7, in depth, in post #437 in this thread.
"...he saved us, not because of righteous things we had done, but because of his mercy. He saved us through the washing of rebirth and renewal/regeneration by the Holy Spirit, whom he poured out on us generously through Jesus Christ our Savior, so that, having been justified by his grace, we might become heirs having the hope of eternal life." Tit 3:3-7
Since regeneration is a term only used here and in Matt 19:28, we have to look at the surrounding scripture/context and parallel passages to understand what it is referring to.
The renewal by the Spirit in Tit 3:3-7 is tied in with the 'washing of rebirth,' the pouring out of the spirit, mercy, and justification by grace.
Justification comes by faith, not before faith (Rom 4:2-10, Rom 4:23-24, Gal 2:16, Acts 13:39, etc.)
The washing of rebirth is a clear allusion to baptism (I Pet 3:21, I Tim 1:5, ph 4:4-6, II Cor 5:14-17, Col 2:11-13, etc.)
Faith precedes rising to a 'new life' in the power of God (Col 2:11-13)
The pouring out of the Spirit also follows faith: (Gal 3:2, Acts 10:45, Rom 5:1-11, etc.)
And note the very next verse! "This is a trustworthy statement; and concerning these things I want you to speak confidently, so that
those who have believed God will be careful to engage in good deeds." The passage is not cutting faith out of the picture at all!
It is merely one description of things that accompany salvation by faith vs. works, as described elsewhere in passages like Eph 2:5-9, Gal 2:15-16, Rom 4:2-7, Rom 15:7-13, Rom 3:21-26, Jd 1:20-21, etc.
Rom 8 also does not support your view that regeneration must precede faith.
First, since Rom 8 begins with 'therefore...' it is important to read Rom 7 for the context to see what the 'therefore' is concluding from.
Rom 7:4-6 "So, my brothers and sisters,
you also died to the law through the body of Christ, that you might belong to another, to him who was raised from the dead, in order that we might bear fruit for God. For when we were in the realm of the flesh, the sinful passions aroused by the law were at work in us, so that we bore fruit for death.
But now, by dying to what once bound us, we have been released from the law so that we serve in the new way of the Spirit, and not in the old way of the written code."
Rom 7:21-25
"So I find this law at work: Although I want to do good, evil is right there with me.
For in my inner being I delight in God’s law; but I see another law at work in me, waging war against the law of my mind and making me a prisoner of the law of sin at work within me. What a wretched man I am! Who will rescue me from this body that is subject to death? T
hanks be to God, who delivers me through Jesus Christ our Lord! So then, I myself in my mind am a slave to God’s law, but in my sinful nature a slave to the law of sin.
Paul is speaking, in context, of how believers serve in the new way of the spirit, and of the struggle between the mind now subject to God and the flesh still corrupted by sin.
"Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death." Rom 8:1-2
Paul is speaking of those in Christ Jesus,
and how the law of the Spirit sets us free from condemnation, even when we still struggle with sin. He is not making any claim about being under the law of the Spirit or needing new life prior to faith. His words are to those in Christ Jesus.
Rom 8:3-8 "For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh, in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit."
Here Paul moves on to the work of Christ on the cross, and specifically how God's condemnation of sin allows us (believers) to now meet the righteous requirement of the law, since believers live according to the Spirit.
Rom 8:4-8 "Those who live according to the flesh have their minds set on what the flesh desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. The mind governed by the flesh is death, but the mind governed by the Spirit is life and peace. The mind governed by the flesh is hostile to God; it does not submit to God’s law, nor can it do so. Those who are in the realm of the flesh cannot please God."
Here Paul is speaking of who our lives are submitted to. Those who live according to the flesh seek to please the flesh, but those governed by the Spirit (believers) bear the fruit of life and peace. If one remains in the realm of the flesh, hostile to God, one cannot please God.
Again, Paul is talking about 'living' submitted to either the flesh or the Spirit, such as described in Gal 5:13-26 as walking by the Spirit vs. giving in to the desires of the flesh. He even warns believers not to submit to the flesh, for it is contrary to the Spirit.
So, how does one stop living according to the flesh and turn to submission to the Spirit? Faith! (I Jn 4:14, Heb 11:6, Jn 3:3-21, etc.)
Rom 8:9-17
"You, however, are not in the realm of the flesh but are in the realm of the Spirit, if indeed the Spirit of God lives in you. And
if anyone does not have the Spirit of Christ, they do not belong to Christ. But if Christ is in you, then even though your body is subject to death because of sin, the Spirit gives life because of righteousness. And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies because of his Spirit who lives in you. Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it.
For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live. For those who are led by the Spirit of God are the children of God. The Spirit you received does not make you slaves, so that you live in fear again; rather, the Spirit you received brought about your adoption to sonship. And by him we cry, “Abba, Father.” The Spirit himself testifies with our spirit that we are God’s children. Now if we are children, then we are heirs—heirs of God and co-heirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory."
Etc. Paul continues speaking to believers of the difference between a life led by the Spirit vs. life by the flesh. Nothing in the context nor word use mandates that a non-believer must be regenerated by the Spirit prior to faith - it is all about how believers, the children of God, walk by the Spirit so we can put to death the misdeeds of the body, and how there is no condemnation for sin for those who have the Spirit of Christ.