I'm no RF guy, so I'm clueless on this stuff. But my understanding is that SDRs aren't radio receivers in the traditional sense. That's the whole "SD" part of it, the device scoops up everything and FFTs it or something.
But they are suffering from the chip shortage. I was interested in using it as an X-band and K-band capable SDR receiver, which is a supported use case according to their media. But we'll see when it ships.
A modern radar gun operates in the Ka-band, Google says between 33.4 and 36 GHz. Assuming 12 bit samples, your proposed solution needs to process a minimum 3.9 gigabits per second, or ~500 megabytes per second. To do this work in software would require a workstation.
Not to mention I'm not even sure if you can operate most ADCs above the 1st Nyquist zone like would be needed here.
There are Radar Detectors that have VG-2 (Radar detector detector) detectors on them.
I haven't heard of a radar detector detector detector detector. Such a thing would be pretty useless for a cop's intent and purposes.