Yes it's a flow problem. Running the pump won't backfluch the inlet regulator unless it's broken. Running the pump will help determine if the regulator is the problem though. If you turn on the pump and get good water flow from your sinks then the problem is at the regulator, not the individual faucets being clogged.
The screen inlet to the regulator may be clogged. If that looks clear then 99% it's the regulator itself.
Good news is those regulator are less than $40(Amazon) and straight forward to replace.