I'd say you should start out with a meaningful, or workable, definition of the word "exist". Whether or not time "exists" is a crucial part of your thinking, but you have offered no workable definition of this word. When you do, I think you'll find that no concepts of the human mind, nor any of the words we use to describe our universe, correspond to something that "exists". Given this, we do the best we can, and one shouldn't get too worried about paradoxes.