Fresh water tank magically fills itself

I "tested" the check valve in the water pump in the following manner: I pressurized the water system (in your case you can just hook up to the city water) then disconnected the water tank to water pump hose (disconnected at the water pump). I then attached a clear plastic hose to the water pump inlet. I actually used the hose commonly used for sucking up the antifreeze when winterizing (in the old days when the winterizing hoses were not plumbed and valved in). I taped the hose in a vertical position then left it for a while. In my case there was some water in this hose so I just marked the level of the water in the hose. When I returned a short time later, the water level had risen several inches and I knew the check valve was leaking. Simple but worked for me. Hope this helps and you don't buy a new water pump if it is not necessary.