DJI, who makes the majority of consumer drones, has already implemented broadcast-based radio ID, but that's not what the FAA proposal suggests and is not the part of the regulation that is being criticized.
In fact, the FAA has explicitly stated that they don't want to use a radio-broadcast system like ADS-B (the equivalent for general aviation). The solution being proposed involves a proprietary internet-based system that has privacy/security implications among other issues -- The system would send the location of any user to a third-party company, would require a subscription, etc.
The solution does very little to alleviate the concerns you are discussing and many more reasonable proposals have been suggested which DO involve radio ID. My favorite solutions involve a combination of a radio beacon broadcasting GPS and registration # in the style of ADS-B for larger drones/aircraft, and for smaller aircraft an app-based system where users "check in" with a flight plan. But keep in mind that the outrage is not over any regulation by the FAA, it's that their proposal completely disregards existing industry standards like those put out by DJI and creates an expensive, hard to enforce system that favors large corporate interests.
Let's keep in mind that there are millions of drones in the US (1.1 million registered with the FAA, and many more that have not been registered because they are <250g or custom-built.) and have not had a single incident involving injury or death. Every single incident involving a drone has involved the concern of violation of someone's privacy or violation of an airspace, and even in those incidents there has been a lot of hysteria involved in the media response. In my opinion this absolutely does not justify preemptively building an expansive surveillance infrastructure around technology that has, so far, proven to be very safe.