Pump cycling as described will destroy your pump in short order.

That's typically why the tank is used, to stretch a nominal number of start-ups over the warranted life span.

But your info is inadequate. If the pump is controlled by the usual pressure differential switch then there must be a leak...in the check valve at the pump or the line coming from it. The check valve can leak internally and/or externally.

Your preference to leave the system on is possible but you either need to supplement with a tank or use a flow control valve.

The pump is probbaly leaking enough water backwards and through the check valves to trigger the pressure switch.

Air is used in the tank to store the pressure, since the smallest amount of water leak wil drop the pressure very quickly.

At the low pressures you have water is not compressible while air is.