I have also been in both SAPR and LSSD.
While some deputies do patrol the county, 95% of them never touch it. The fact of the matter is that most of the time, too many calls and situations in the city lead to no available deputies to patrol the city, let alone the county. No matter what happens in a "perfect world," that is the true reality of the situation.
SAPR to SAHP is a good call, it should be done. They do it anyway. They're good people too.