Well done; that is exactly correct. As we want to model the systems more accurately the computational resources get insane.

If time is not continuous, then it would be possible to use a discrete model to predict these chaotic systems arbitrarily far into the future in our universe.

Not necessarily; in fact (in my opinion) all that would do would set a lower bound on the time step required in your numerical simulation to achieve a "perfect" model. You would still have to solve the equations over and over again to know the future state.

Quantum computers may be able to help us solve certain classes of problems much more efficiently. But even these don't change the fundamental nature of reality; there are still unknowable future states.

It also doesn't matter if time is discrete or continuous; since we live inside time, we cannot experience the difference. The universe could run for a second, then stop for a year and then run again for a second, we would experience it as two continuous seconds.

Maybe in the future we will work out some way to step outside the normal flow of time and answer that question.