Or do what the Japanese do: every bike has a serial number etched into the frame and is registered to you (similar to a car). If your bike is ever stolen, the police can trivially find it by the serial number and return it.
I believe every modern bike has a unique frame number stamped. If the police recovered a known-stolen bike it could be returned. However, buyers and bike shops don't make a habit of running frame numbers past the theft database, so it's mostly useless.
Every car has a serial number in several places, in theory it should be trivial to find a stolen one, yet plenty of cars are stolen each yer. Clearly this does not solve the problem.
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.
This is another example of something that works great in Japan and wouldn't work at all elsewhere. In other countries (esp. anywhere in the Americas), this would never work, because there's no shortage of someone willing to buy something that may be stolen.
