Major problem your claim requires they have life before believing when the verse puts believing before lifeJn 20:31 is for the regenerate sheep, that they may have life Jn 10:10
10 The thief cometh not, but for to steal, and to kill, and to destroy: I am come that they[The sheep] might have life, and that they might have it more abundantly.
Can you deal with the fact that life was received upon believing, and thus the previous condition must have been one of spiritual death. But One who has been regenerated is not in a state of spiritual death but of life
John 20:31 (LEB) — 31 but these things are recorded in order that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.
BTW John 10:10 in itself says nothing regarding the order of faith and life
context however
John 10:4–9 (LEB) — 4 Whenever he sends out all his own, he goes before them, and the sheep follow him because they know his voice. 5 And they will never follow a stranger, but will flee from him, because they do not know the voice of strangers.” 6 Jesus told them this parable, but they did not understand what it was that he was saying to them. 7 Then Jesus said to them again, “Truly, truly, I say to you, I am the door of the sheep. 8 All those who came before me are thieves and robbers, but the sheep do not listen to them. 9 I am the door. If anyone enters through me, he will be saved, and will come in and will go out and will find pasture.
is contrary to your theology as you posit sheep who do nor know his voice, follow strangers and do not flee from them but listen to them