I agree, to "see" is to experience, and once again in John 3:3 birth precedes "seeing" or experiencing.
Other Scriptures
Several texts from 1 John demonstrate that regeneration precedes faith. The texts are as follows: “If you know that he is righteous, you may be sure that everyone who practices righteousness has been born of him” (1 John 2:29). “No one born of God makes a practice of sinning, for God's seed abides in him, and he cannot keep on sinning because he has been born of God” (1 John 3:9). “Beloved, let us love one another, for love is from God, and whoever loves has been born of God and knows God” (1 John 4:7). “Everyone who believes that Jesus is the Christ has been born of God, and everyone who loves the Father loves whomever has been born of him” (1 John 5:1).
We can make two observations from these texts. First, in every instance the verb “born” (gennaô) is in the perfect tense, denoting an action that precedes the human actions of practicing righteousness, avoiding sin, loving, or believing.
Second, no evangelical would say that before we are born again we must practice righteousness, for such a view would teach works-righteousness. Nor would we say that first we avoid sinning, and then are born of God, for such a view would suggest that human works cause us to be born of God. Nor would we say that first we show great love for God, and then he causes us to be born again. No, it is clear that practicing righteousness, avoiding sin, and loving are all the consequences or results of the new birth. But if this is the case, then we must interpret 1 John 5:1 in the same way, for the structure of the verse is the same as we find in the texts about practicing righteousness (1 John 2:29), avoiding sin (1 John 3:9), and loving God (1 John 4:7). It follows, then, that 1 John 5:1 teaches that first God grants us new life and then we believe Jesus is the Christ.
You conveniently overlook several verses in John 1 chapter 5.
1 John 5:4 For whatsoever is born of God overcometh the world: and this is the victory that overcometh the world, even our faith.
5 Who is he that overcometh the world, but he that believeth that Jesus is the Son of God?
It is true that whoever is born of God has the capacity to overcome the world (sin, work acts of righteousness), but notice vs. 5 says also that the person who believes overcomes the world.
So, believing cannot be the cause of that power (regeneration) that enables us to overcome the world and be the effect also.
1 John 5:1 Whosoever believeth that Jesus is the Christ is born of God: and every one that loveth him that begat loveth him also that is begotten of him.