No, I did not make your point and that you think so is scary. No, man is NOT choosing there. The "him that cometh to me" are only those the Father gave to Christ and no one else.
If God hasn't specifically given someone to Christ, then they cannot/will not come to Christ and therefore not seek God.
How could you possibly miss that? Did you look only at the last part of the verse and not the first part? It's not that difficult to understand if you read the whole thing. How is it possible that you can't understand it?
Here look at it again.
Can you see the "giveth me", the "shall come to me", and the "that cometh to me"? They are all linked to each other and complete each other.
[Jhn 6:37 KJV] 37 All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out.
Read these verses too. They clearly tell us that natural (unsaved) man will NOT seek God.
[Rom 3:11-12 KJV] 11 There is none that understandeth, there is none that seeketh after God.
12 They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.