This is exactly what is done by the comm centers that I have worked on. All radio and phone traffic is recorded and can be accessed through an official request. I believe that any citizen can request a recording through FOIA. Recordings may be redacted for privacy or security reasons.
If the NYPD would just go P-25 trunked on their whole system they wouldn’t have to worry about whackers QRMing them with BoJunks and wouldn’t need to go big E. Now if they don’t want folks monitoring their traffic via cellphone apps or scanners, well that is a whole different animal all together which, yes going E would put a stop to. I however enjoyed many years listening to them on during my youth on their UHF frequencies with my old Radio Shack scanners. I still miss hearing that priority signal tone when certain calls would go out…. I remember when there was 3 separate departments NYPD, Housing Police and Transit Police and buying the Radio Shack Police Call frequency guide every year and wondering what color the cover would be that year. If the NYPD ever goes full encryption, I feel it would be an end of era for sure.