I don’t know if I ever thought about it this way, but here goes:
Logic is actually really important to theology. The material sciences can rely on empiricism to some degree; they probably tolerate fuzzy logic, false identities, and a total lack of critical thinking pretty well if observational data gets them back to reality. Theology lacks much empirical data, so it largely lives on logic.
Ganesha’s existence isn’t illogical. Neither is transubstantiation. Those may be untrue. The outsider may find them fantastic, incredible, and head-shakingly weird. But to call them illogical is a failure to understand critical distinctions. The logical is not the same as the real. Something can be entirely unreal and completely internally consistent.