I’m currently in a similar situation as the OP, and am benefiting from “two wrongs don’t make a right—but three lefts do!”
I’ve got both the problem of city water connect is providing good water pressure as indicated by my adjustable water regulator but doesn’t translate into good pressure inside the rig & a slightly leaking backstop valve inside my water pump.
Typically, I benefit from turning on the pump even when connected to city water. having an empty tank means turning on the pump won’t help. But taking additional time to fill the 50 gallon gravity tank is both an added hassle, and gets many quizzical looks/comments from friends and neighbors about filling the tank when there’s a water connection at my site. Luckily, there’s a leak from the city water piping inside my rig that backfills my tank.
So I usually just sit back and leave the pump on. At some point, enough water is used and the reverse leak inside my pump fillis up my tank. When I see water slowly spilling out of the overfill outlet of the tank, I turn off the pump for a while. With a family of five, lots of friends with kids and dogs, and an outdoor kitchen—enough water gets used that I can turn on the pump again for when me and the missus need it for our showers.