The problem was simple. You went around the problem, which is OK. 99% of the time when a water pump surges or pulsates, the WATER SAVERS in the faucet necks cause a restriction that interrupts the psi flow of the water pump. You remove the neck screw on screen and pop out the plastic water restrictors. On a Shower head(the cheaper ones) you may not have this option and either replace the shower head or drill out the small hole in the neck to allow more water flow. Always open 2 or more faucets when this happens to determine if you have a water pump problem. Usually just opening both hot and cold on one faucet will stop the pulsating and that narrows it down to the restriction. Doug
PS, it would be rare for any air in the system to cause water pump pulsating/cycling. Doug