I downloaded the fully unlocked save, but when I go to items, it shows that there are a lot missing. I disabled steam cloud and made sure there was a save in each slot. Am I doing something wrong?

Everything is all right, these savefiles weren't unlocked manually, just hacked. That's why there are a lot of items missing in collection page.

So if I were to submit a run using this save, I'd be alright?

Yes. The items are unlocked.