"How can a black hole exist if we don't see anything ever cross the event horizon?" is probably a better question.
Seems to me that the question is asking how do we know
if the black hole exists if we can't see beyond the event horizon, since we
implies our perspective. Reminds me a little of that old phrase, if a tree falls in the woods by itself does it make any sound?
In terms of direct observation the singularity itself can never be witnessed because of the properties of the event horizon, but the very fact that the event horizon exists at all means something is inside there doing terrible things with space and time.
Mathematics settles the rest; one you have an accurate figure of the mass and width of the object, math can prove the object cannot even resist its own titanic gravity well. Past a certain threshold, it can be nothing but a black hole.