Why is irreversibility a bad thing? Is it just the loss of useful state which reduces the internal entropy?
Is irreversibility ever useful? Wouldn't some amount of discarded bits make it harder for an adversary to infer the state?
If something is irreversible then it's not 1 to 1 and you don't use the whole state space, so you'll likely have short cycles.
Second part, again yes. But you can add that useful irreversible transform in the finalization of the output, and not in the state transform itself.
I'm self taught so my understanding and terminology may not mesh with the people who have been taught this.
You mentioned the lack of cryptanalysis; there is also the lack of rounds (which prevents researchers from breaking partial versions to ease their study, and allows setting a security margin).