Well that's what they are supposed to do. That is until they don't. I think the pumps used by Airstream and probably by most of the RV industry are reliable and long lived but they will sometimes loose their ability to pull water up from the tank to prime the system. This requires the pump being capable of pulling a vacuum enough to draw water from the supply and push it into the system enough to push the air out and keep the plumbing system pressurized. In my case the motor ran well but lost the ability to pull the vacuum and the water. SO, the remedy here is a re-build kit from SHURflo. It consists of a replacement head for the pump and an "O ring". Replacement is simple and only requires a philips head screwdriver. These kits are available on E-Bay for about $14 + shipping. The replacement may vary based on model # so make sure you get the one for your pump model. Mine consisted of a head with 4 small diaphragms. These can fail for a number of reasons most of which you won't be able to detect visibly so go ahead and just replace it.
I had purchased a new pump since I was on the road when it was found to not work. Since the motor worked I kept the old pump and re-built it later. Now, more recently a different pump related issue developed giving me the opportunity to install the re-conditioned one. I was at a campground while on the Airstream Cajun Country caravan back in March. We were hooked to city water. I noticed water coming out of the fresh water overflow tube and puddling below. There must be a check valve in the system to prevent city water from flowing into the storage tank but I had no idea where to find it. One of the Airstreamers with us on the caravan happened to have been an AS tech in his prior life and proved to be a wealth of information during the caravan so to him I went with my inquiry. Well, unbeknownst to me, the check valve is built into the pump and although the pump was working and doing it's pumping function OK something had caused the check valve to fail and allow water to back flow into the tank causing my overflow. So it is a simple matter to change the pump so I put the re-built one on and problem solved. Since returning home I've obtained new rebuild kits and will take care of it ASAP. But prior to just replacing the head in this one I think I'll take it apart and flush the diaphragm head in hopes that it is a simple case of some debris causing the check valve failure. It that doesn't work I have the kit to install.
None of this will work if the motor goes dead. I've never heard of this but it is, of course, possible. I don't know if the internals are accessable or re-buildable even if they are so you're on your own should the motor fail.
So, should your water pump fail to prime itself or maintain pressure in your plumbing system re-build it if you have the opportunity. If you choose to replace the pump save the old one and re-build it later as a backup. In our trailer the pump is located below the wardrobe closet, curb side. It took me longer to empty the floor of the closet than it takes to replace the pump.
See ya on the road sometime.