Currently I don't think you really know what is wrong with the thing. My guess is it isn't the unloader and it could be a bad starter or run capacitor. If it really is the unloader then you won't hear it vent when the pump shuts off. It should start just fine if you wait long enough for the pressure to bleed off naturally (say a few hours).
Does your compressor hold air for at least a few days when not used? It will make things easier to figure out if we at least know that.
Here is how you can check. Run it until it shuts off. Look at where the pressure switch actually pushes on the unloader valve. Did it move the valve? Did you hear the air rush out when the pump turned off? If yes, then your unloader valve is good. A second way to check is fill the thing up then leave it alone for a few hours. During that time the pressure in the line from the pump to the check valve going into the tank should bleed off. See if it will start (you may have to bleed out some tank pressure to get it low enough).
Looking at the pressure switch...
Do you know if the pressure switch actually turns on/off when it should? I'm guessing that is something like a 120psi max compressor. Fill it up then unplug it when the motor turns off. Leave the switch on. Now when the pressure bleeds down to the lower pressure limit (probably 100 psi) you should hear the switch click. You should also feel that when the pressure is high (above the lower limit) the switch doesn't have much resistance. When the pressure is low the difference between on and off will be bigger. If you bleed the tank pressure down and you don't hear/feel the switch click to turn the motor on then you may have a bad switch. If you do feel it but the click over pressure is say 60 psi then you know the switch is out of adjustment/bad, Note these switches generally can be adjusted so you may just need to set the cut in/cut out pressure correctly.
If you have a bad run capacitor the motor will struggle to build full pressure. It doesn't sound like that is your issue. If you have a bad starter capacitor then you may have issues starting because the motor isn't getting up to speed before the pressure in the head is too much to handle at low RPM.
Do you know if the pump is running at it's rated speed? Time how long it takes to go from empty to full. For example this one:
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This calculator says an 8 gallon compressor filled with a 4 cfm pump will take about 2:11 (min:sec) to go from empty to 120psi. If your compressor takes a lot longer than the estimate then it could be a motor or pump issue.