After Avengers: Endgame, the Infinity Stones are destroyed and therefore no longer exist in the universe. How does that fit into the Ancient One's explanation that the stones MUST exist in the universe without it falling apart?
We don't know, in fact things might not even be different. All we know is that The Ancient One tells us that the Infinity Stones control the flow of time and removing one spawns an alternate reality. Though we are never told that the Infinity Stones must exist in the universe otherwise it will fall apart, just that they help control the flow of time. This leads us to three possible theories of what can happen post Avengers: Endgame:
Time is unregulated: Time is now unregulated and so any number of alternate realities can be created by using the time travel and altering things.
Time is strictly set on the "main" timeline: There are no Infinity Stones left and so you can't create an alternate reality.
There are 7 realities: the main reality plus six realities where each Infinity Stone was removed (destroyed).
I'm of the opinion that it is probably number 2 as the Infinity Stones were destroyed at the start of the film and so we saw what is apparently only one reality with no others created. However, it is all up in the air until Marvel decide what they are going to do post Endgame and in fact it might play a role in GotG3.
As Josh commented on the SFF question it's also unclear if the Stones were destroyed completely or just reduced to their atoms. If the latter they are still around and everything still works out as The Ancient One expects.