Water heater Bypass should be shut for normal use. Not sure why they have everything open. If the inlet and outlet valves were open during the winterizing process then there would be a lot of pink stuff in the heater. If they opened it afterwards then there would only be a little until (if you listened to them) you run the pump and flushed a bunch in. No matter what, things are messed up. First of all I would close the inlet and outlet valves and flush the system using the city water hookup not the on board tank and pump. Open everything, sinks , shower, outside spray, and under coach drains. You should run the pump just to clear the pump lines. The filter housing (filter should have been removed during winterizing) should be flushed with the city water supply unless it is bypassed when on the city water supply then the pump would flush it.
Since it is hard to flush out the water heater if pink stuff got into it, I would pull the drain plug to drain anything out. With the plug out you could open the inlet valve and flush it using the city water supply
Once done you can reinstall the drain plug make sure the inlet and outlet valves are open and the bypass is closed. Reinstall the water filter and you should be good to go.