It solves the problem in Japan. Japan just isn't like other countries; they have citizens who actually care about their society and don't want to be seen as leeches on it.
It's no wonder they don't want to import a bunch of foreigners. I can just imagine how their society would be completely ruined if they allowed millions of Americans to move in. Bike theft would be the least of their problems.
The question is - would japan have a bike theft problem without bike registration scheme? I suspect not, for the reasons you mentioned. In which case, the registration numbers are more of a curiosity than anything.
My own country(Poland) had mandatory bike registration 30 years ago(during communist rule) and no, it didn't stop bike theft.
It's no wonder they don't want to import a bunch of foreigners. I can just imagine how their society would be completely ruined if they allowed millions of Americans to move in. Bike theft would be the least of their problems.