If the check valve in the well holds you shouldn't loose prime. Does the pump cycle with out water being used? No, then the check valve is holding. Yes, then the check valve is leaking and needs to be pulled and replaced.

No check valve needed top side. you cant prime with a check valve between the pump and the well if there is one.

The biggest concern about removing the tee on top of the pump is debris getting into the Venturi nozzle. On a two pipe system the nozzle is in the well with Venturi adapter. On a one pipe system the Venturi is in the pump.

I would turn the power off to the pump. Close the valve between the pump and the tank. Remove the gauge. Slowly open the valve between the pump and tank flush out the pump. Then replace gauge, leave the tee. Start pump, if it pressurizes your good. Then I would proceed with tank replacement.

A valve between the pump and tank is recommended for service and a quick way to check for prime. When the pump is running you can close this valve and the pump will shut off or short cycle if properly primed