"However, a pulsating flow at the kitchen sink doesn't fit a leaking check valve. Nor does the water flow stopping with a hose valve open."

If the CV has failed, then it could allow pressure to fall rapidly once the pump cuts off and have a pulsing effect. The same could be true of the water hose. Hard to say when we aren't there to actually see these things. However, it is well worth the effort to double-check the pressure tank since that would be the easier fix.

Mr, do let us know how it all works out. Your case is a little hard to nail down.