After replacing pressure tank water stops completely for several minutes at a time

Pressure tanks don't develop pressure they hold pressure and the air chamber is there to push the water out of the tank that is stored there reducing the number of on off cycles on the pump providing a more uniform pressure at the taps. It sounds like your pump may be the weak link if your pressure switch is turning on prior to running out of water and it still takes 3-4 minutes to get water, the check valve or foot valve in the pump may be leaking and it takes that long to refill the pipe to the tank, I have seen this more often with jet pumps than submersible pumps but it can happen to both. First verify the pressure switch is turning on where you expect, if it is I believe your next step is checking for a bad check valve/ foot valve if your flow is good once water makes it to the tap.