One of the key reasons airlines don’t allow passengers to stay on board, even if the flight is a through flight with the same equipment (which, as noted, is quite rare in non-Southwest operations) is that, legally, they can’t (well, they could, but it would be complicated/expensive).
The FAA requires one flight attendant for every 50 seats (not passengers, but actual seats fitted on the plane). The required FA crew must be on board (meaning on the plane, not in the gate area, not even on the jetway next to the boarding door) before a single passenger can be on the plane.
Since virtually all non-Southwest flights in the US serve an airline’s hub (or focus city/mini-hub), there is a high likelihood that, even if the plane is continuing, the crew isn’t. Therefore, once the last passenger is off the plane, the FAs leave. The next crew doesn’t board the plane until shortly before boarding. Even if the crew is continuing on, if even one flight attendant had to step off the plane for any reason (assuming they are staffed at the legal minimum to begin with), the entire planeload of passengers would have to leave first. Therefore, it’s much easier to just make everyone get off and get back on as a matter of policy.