Never have heard of 4 bad pumps! You can buy a rebuild kit.
When on city water connection, do you run the pump?? The answer should be no. I have left mine on for a day or so, running dry and have never had any issues. I have heard of others who have left them on for a week running dry and no issues.
I think it is time to do some serious diagnostic work. My guess would be your pickup tube, is cut very square and it vibrates down to the bottom of the tank while traveling and cannot pick up. However, that does not explain not being able to suck out of the winterizing port.
I will assume your TT is similar to my HTT in how it is plumbed. At the city water connection if you remove the rubber filter washer, there is a one way valve right behind itto ensure water does not exit the TT when using the pump. Remove the washer and use a screw driver to push the valve in a little bit to make sure it works. Then I would check the filter screen right before the pump. You should be able to unscrew it without tools, remove it, check the screen for debris, clean and reinstall. If the clear housing is foggy or full of fine looking fractures replace it. Air will leak in (I had this happen to me) and you lose prime and pressure. Next I would check your winterizing bypass valves at the pump. I have never tried to take one apart, or even if you can, if not; open and close them a few times. Iím suspecting that one of them may not be closing all the way, there might be construction debris such as pipe tape or plastic stuck in the valve keeping it from closing all the way, or damaged the plastic ball, allowing air to pass by. Next lightly tug on the pickup tube (FW tank), I am not sure how it goes into the tank, maybe the hose was cut perfectly square and it vibrates down and becomes blocked as it is bottoming out on the tank. If none above works, remove the pump, on a table outside, setup the pump, with your 12V battery and some hose and bench test it. Can you pump water from a 5 gallon bucket? If so, then the issue is not the pump. If the pump does not work, you can buy a rebuild kit. I would buy the kit, carefully open the pump, and examine what appears to be wrong, rebuild it. A rebuild kit would be a lot cheaper than a new pump.
Good luck, this sounds like a unique issue, could happen with any TT from any vendor.