The toilet water valve and the toilet sprayer are full flow devices.....therefore the pump can not reach 'stop' pressure set point.
The faucets etc do not allow for full flow......pump reaches 'stop' pressure set point...momentarily stops then restarts when pressure drops. So pump is cycling ON/OFF. Very typical on demand pump operation when low flow demand occurs.
Remove restricters in hot/cold water lines to faucet, in faucet and in hose to shower head....higher demand/less pump cycling
You can also increase 'stop' pressure setting.......on end of pump head is an adjustment screw. Clockwise to increase....but only make small adjustments and never more than 1 1/2 full turns. The pump should operate for 1 sec. with at least 2 sec. "OFF time"
Adding an accumulator tank in pump discharge line will also help smooth out pump cycling
Hmm, I think it's the opposite. When he's using a higher gpm, the pump runs constantly. At lower flows, it pulses to try to match the flow. What he's describing sounds perfectly normal for any pump without an accumulator tank.
You can install a small captive air pressure tank to even things out if you want, but I think your system is running as it was designed to.