This would be a technical nightmare given how police radios work. However, you can submit a FOIA request for radio recordings, which basically does what you sug

This would be a technical nightmare given how police radios work. However, you can submit a FOIA request for radio recordings, which basically does what you suggest but with a longer delay. The radios do support over-the-air keying, but it is rarely used. Having physical radios come back to the shop for key refreshes provides positive control over radios, and prevents a stolen radio (which happens more often than you'd think) from listening in on a SWAT or drug operation.