LASD is a strange agency. I often wondered this as well before living somewhere close enough to scan them.
Each area has two frequencies, a Dispatch channel and an L-TAC (local area tactical). The dispatch frequency has the busy tone but the L-TAC doesn't.
The PSAP (911 call taker) is located at each local sheriff station. The local station dispatcher answers 911 and enters call information into their CAD system.
That CAD info then pops up on another dispatcher's screen at the consolidated Sheriff Communications Center (SCC), who voices the call over the dispatch frequency.
Once the unit is responding to the call, they often switch to L-TAC to coordinate with other responding units "Let's 911b here, I have a tazer", etc. Conveniently, the original station dispatcher is also on L-TAC and assigned a unit ID (for example, Palmdale station dispatch is "260D", Lancaster is "110D", etc). So if the responding units need dispatch to call back "the informant" to get more info on a call, they ask the station dispatcher to do this over L-TAC, not the SCC dispatcher who voiced the call.
If there's a larger incident requiring officer to officer or officer to airship coordination (like a large perimeter, pursuit, foot pursuit) SCC can activate "the patch" on the dispatch channel, which turns repeat mode on and drops the busy signal.
It's a pretty bizarre setup that virtually no other law enforcement agency uses in modern times, but they seem to make the best of it and I haven't really heard any officer safety compromised because of it. When you factor in how SCC uses "dispatcher trunking" to allow for lower staffing levels (their computer system automatically adds and removes dispatch channels to a dispatcher's console based on their work load), it makes sense why they've been so against abaondoning it. Imagine being a deputy assigned to Lancaster and hearing units responding code 3 and wondering if you should be enroute to their call, only to then find out they are in North Hollywood. Then a minute later, you don't hear the North Hollywood units anymore. That would get confusing and annoying. But that's more the fault of the "dispatcher trunking" concept, not the repeater. I'll bet it will still probably be easier to hire more SCC dispatchers and turn busy tone off than it will be to jury rig the busy tone onto LA-RICS.