Just in order that I understand you here, in the meaning and use of the word "see" in John 3:3 does not really affirm total depravity and regeneration coming before faith. I have often read and heard from the Reformed community that it does.
One verse never affirms anything. And speaking for myself as I am the only one I can speak for and my view of Reformed theology, one verse
is never used to affirm or deny anything, or establish a doctrine. The entire teaching of the Bible is used. For the sake of space and time, singular verses are often given on a forum to show that what is being said is indeed said in the scriptures. And then, unless the discussion immediately digresses to he said, she said, defenses such as "That's Calvinism" or "That is a false teaching" are used as refuting what is said, the conversation can move forward so that the whole counsel of God on the subject, and His self revelation can come into play.
That said let's go to John 3. In verses 1-2 Nicodemus came to Jesus by night, acknowledging that He must a man of God, and he did not even ask a question at that point. But he must have had one that Jesus surmised for that is when Jesus said, "Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God." The word "see" has many usages. It can mean to perceive by the eye; to imagine a possibility; to understand; to come to know as in discover; to have experience of i.e. undergo to name a few. But of course that was not the end of the conversation.
Nicodemus took Him literally and asked, "How can a man be born when he is old? Can he enter a second time into his mother's womb?"
Jesus responds by saying, "Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God." This is likely a reference to OT passages that link "water" and "Spirit" as a single spiritual truth to express the pouring out of God's Spirit in the end times (after the resurrection) and the purification and new life that flow from His arrival. (Is 32:15; 44:3; Ezek 36:25-27)
In any case, Jesus is saying this new birth which is new life, new birth comes about through the agent of the Holy Spirit, and that unless this occurs one cannot enter the kingdom of God. Therefore the "see" in verse 3 relates to "cannot enter" in this verse, and "see" is logically interpreted to mean "enter." That is, experience or undergo.
Jesus goes on to underscore that this new birth is of the Holy Spirit and not through any actions or will of a man, such as believing. The new birth must come prior to believing.
Verse 8. "Do not marvel that I said to you, 'You must be born again' The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit."
And if that were not enough it has already been attested to by John in John 1:12-13 "But to all who did receive Him, who believed in His name, He gave the right to become children of God who were born, not of blood, nor of the will of the flesh no of the will of man, but of God."