What worries me, however, is whether such a manual override would lead to more or less problems
The eternal argument not to act.
Fact is that on the other fly-by-wire product, there has not been one incident where a manual override of a computer has led to a worse situation. (Not that i would know of an incident where computers gave erroneous signals, though.)
Fact is as well, that in at least one Airbus (QF72) such erroneous signals led to an erroneous input that was at first not recuperable by the pilot. (A override capability would have been very welcome I suppose.)
Reread the two facts and then the entry argument and very basic common sense will give you an answer, at least to me ......