Respectfully, I am not chasing down and defending every OT passage that anyone has ever told you refers to Jesus. Nothing personal, but that's not my job and too darn much work. It is also pointless and has the whole thing backwards.
I do not believe in Jesus "because the Bible says so". I believe in Jesus because he has proven himself real to me. If and How God and Jesus reveal themself to you, is between you and Him. I can convince you of nothing.
Starting from what I know to be true about Jesus/God, I find that the Bible (NT) has the best track record at explaining and aligning with reality as I perceive it. So I believe the Bible because Jesus says so, and not the other way around.
To address the issue of saved and the OT vs NT from the 10,000 foot view one last time. Other parts of the Bible tell us that it was Satan, but I say screw the snake, it is not the important detail. The important detail is that starting with even the first 'perfect' people, we are in trouble, God promised to send someone to fix it and people have always either trusted God, or not. Those who trust God, have that trust credited as righteousness. Jesus fixed my sin, changed my inner being, and transformed my life.
I believe it was the same for Seth and Enoch and Noah and Abraham. You are free to believe whatever you want.
If you really want to argue about literal snakes, then did Cain, Able or Seth crush the head of the talking snake and hurt their literal foot? Why did God make such a big deal about an event so trivial that it wasn't even recorded?
... unless it was a prophecy about a coming Son of Eve who would remove the curse that led to death for the human race. Then everyone who trusts in God, whether they know it or not, it trusting in Jesus ... the one who fulfills that promise.
I don't know but that sounds more backwards. How does anyone know of Jesus without the NT? With only the Jewish Bible how does anyone know of Jesus at all? The Messiah was to do certain things. Did Jesus do all of them?
Some Christians push the literal interpretation to the limit. Like a young Earth, a world-wide flood, and a walking, talking serpent that was really Satan... that before he was known as Satan, he was Lucifer. Where do Christians come up with all of this? From the Bible and, I believe, a little influence from Greek and Roman and other pagan beliefs. I don't believe it literal at all.
But I do believe that people that put their trust in a "higher" power or some other spiritual kind of belief do get an inner strength and a sort of confirmation of their beliefs. But it happens in all religions. Even within Christianity a Catholic gets answers from praying to Mary. People get comfort from calling on the name of Jesus. But was "Jesus" his real name? No, it was some Hebrew or Aramaic name probably, but they say "Jesus" and believe and it works. Some Hindu calls on the name of Shiva or maybe Krishna, and it works. Does it work beyond this life? I don't know. As far as I know, it wasn't that big a deal or talked about that much in Judaism, but it is everything in Christianity. Where is the consistency of "truth"? God changes his message and focus on what is important.
And, speaking of consistency, this whole thing that "sin" led to death is a Christian thing. The serpent is cursed, but you don't care about any of the curses but the one that you can use to connect to Jesus. Then, Adam and Eve received curses. None said that now they will not live forever but will die.. To me, it is way more likely that living and dying was always part of the cycle. Things rust, crumble, and slowly fade away. You need this interpretation that death entered the world because of sin to make Jesus' life and death have meaning. Where do you get that? Did God reveal it to you? Or, did you read it in the NT?
Like you said, you can "believe what you want", and so can you and everybody else. And only a few believe your form of Christianity. How many of the other, non-tulipers, do you consider "saved"? At what point do they believe in enough correct stuff about Jesus to be "saved", and some one else believe in too much wrong stuff that their faith and trust is considered to be in a false Jesus? In that case, aren't the details of what you believe important?