The airside connection all terminals are still relatively new, so it is understandable not everyone may be aware. Yes, even for airline agents - they generally have no need to venture to other terminals.

I have done the United/other airlines on the south side to Delta/Sourhwest transfer, multiple times, at LAX. Assuming you have TSA Pre, it is generally faster to do this via landside: T7 (landside) to T6 (landside), and walk across East Way, and make a left turn once you are on the other side.