But that phrase isn't grammatical without "that"!

"All great storytellers" starts a subordinate clause.

Yes, in some cases, you can omit the "that" for short subordinate clauses in English.

I don't believe that this is the case here, but even if it is, "that" makes the sentence clearer by signalling a dramatic change to the sentence structure - that there's another little sentence embedded in it.

I might add that the original first sentence has another grammatical error relating to that! "All good writing brings escape of some sort, that makes time disappear..."

Again, "that" introduces a subordinate clause, but grammatically, it's attached to "sort" - but the meaning makes it clear that you want to attach it to "escape".

There are various ways to fix the sentence, and in this case, perhaps the best is to add a little redundancy.

I used to have a horror of any repetition - but then I learned Indonesian that is less picky about it, and I realized it made for more elegant sentences in some cases.

"All good writing brings escape of some sort, *an escape* that makes time disappear" would be my suggestion.

Removing excess words can be good. But if you end up with a non-grammatical or unclear sentence, no.