Governments probably wouldn't care about hobbyists, but they would care about all the commercial enterprises that already adopted drones. Delivery drones and light shows are the most well-known, but they're also used in weather monitoring and agriculture.

Maybe this is the time for drone transponder signals? Require every drone handler to be registered, then have the drone broadcast some signal with the registration ID + some cryptographic hash?

Then shoot down everything that has no transponder or a blacklisted ID.

Yes, this will encourage ID theft (or just theft of the entire drone), but even then, a stolen ID could at least be a starting point for an investigation.