3gpm is about half of what a typical tub/shower valve can supply, given 'normal' pressure and 1/2" pipes. It would take a LONG time ot fill a tub at that rate! Is this the only bathroom with that problem (i.e., do you get good flow and pressure at other places in the house)? Some valves have screen filters on the inlets, and those may be clogged. Since taking the cart out doesn't improve things much, it is likely the supply, and not the valve.
Is it the same on both hot and cold, or is the cold volume much better? When they replaced the WH, they had to turn the water supply off. It's possible that valve is not fully opened. If it was a gate valve, it may be broken, and the flow you are getting is that which can 'slip' by. If it is a broken gate valve, the handle will (usually) just keep turning, and never stop at the one or off position.