Renormalization is well established theory. It is observed in nature in the form of scale invariance during phase transitions. The property of renormalizability is something to be expected in physical theories. If a theory was not renormalizable, that would be an indication that something may be wrong.

You may take issue with the fact that infinities arise in these equations but these infinities are not the same type of infinities that you are used to. They are subtracted off in a way that is understood and mathematically rigorous. A lot of the quotes you list in your link are from the 70's. Since then, a lot of work has been done in this area and it is a widely accepted technique in quantum field theory. In my mind, they are not so bad once you get used to them. The infinities do not actually arise in the physics but in the mathematical tools used to derive the physics. It is sort of like you have an infinite sum and when you look at individual terms you might see infinite. But the individual terms by themselves are meaningless. To get the answer you have to add all the terms. When you do this, you see exact cancellation between the infinities and therefore the final sum is something finite. You can see why this would make mathematicians uneasy, but it is a completely rigorous and well established technique.

But your argument makes no sense. It amounts to saying, here is a theory that I don't understand, maybe there are other things I won't understand, and maybe one of those things will be proof of god. If you understood renormalization, you would not think it is proof of god and the same would go for any future theories we may not understand. Finding something in nature we can't explain doesn't mean god, it just means we haven't understood it yet. Currently, there is not a single observation in nature that we have not been able to describe using basic physical principles. Let that soak in a bit. So what makes you think in the future there will be an observation we will not be able to explain?

Also, you do not understand the concept of hidden variables. These rule out any hidden complexity on the quantum realm. If there was any type of order or structure at all, it would break the inequality. This has already been ruled out experimentally.