If the motor turns on then the pressure switch is operating as it should. The act of changing the switch must temporarily "fix" whatever is wrong. That "slight leak" will likely cause the pump to lose prime after a certain amount of air builds up from the bubbles. You prime the pump to get the flow started but over the next several minutes the leak causes a loss of prime and water flow stops.
I'm a bit confused when you say you didn't use the pump switch. Is this water pump perhaps NOT in an RV water system? Cause RV water pumps have an integrated pressure switch so there is no way to avoid using it. Is this pump in a fixed home or similar application? We do RVs here...
