While there are some similarities to the ham world, there are some major differences that must be considered.
My ham radio is capable of emitting 100 watts of radio frequency (RF) power that is capable of traveling around the world, potentially interfering with other radio operators around the world.
Yet, many millions of people use RF emitting devices -- Cell phones, CB radio, FRS, GMRS, WiFi, bluetooth, microwave ovens, etc., without requiring registration of any type.
Imagine requiring every cell phone or WiFi user having to register (i.e., get permission from the federal government) because some nimrods were playing with GPS jammers near an airport.
In the RF world, there are various levels of restrictions based on the frequency on which you operate, as well as the power output (e.g, the range over which you might have an impact). For the vast majority of RF users -- cell phones, WiFi, FRS, etc. -- operation is permitted without any license and registration (with a few notable restrictions about WHERE you may operate these devices, such as in hospitals, commercial airlines, in the vicinity of radio astronomy observatories, etc.). But, if I step up to power output and frequency ranges that might affect a broad area of users, then licensing is required.
Accordingly, there are various levels of licensing for RF users -- several levels of ham radio (hobby) licensing that grant different levels of frequency, mode, and power privileges, as well as marine, aviation, and commercial users, radar systems, etc.
If you want to use the ham radio analogy, there should be no registration for flyers (of any type) as long as you meet certain operational requirements: Altitude, range, proximity to airfields, etc. For the vast majority of r/c flyers, this means that we can operate as before without the FAA having any interest in our activities.
However, there could be levels of certification for those interested in long-range FPV, high altitude flying, commercial operations, etc. In other words, if your activities could conceivably impact other users of the National Airspace System (or the general public on the ground), then you would be required to prove some level of competence, safety, and awareness of appropriate regulations.
And, of course, severe punishments should be meted out for those nimrods who fly their off-the-shelf toys in such a way as to endanger or impede the general public.