You keep making assumptions, the pump may be partly OK, normally the switch light flashes when every thing is locked securely with no faults but in your case as the doors are not locked when it flashes you have a problem.
Just because the lights flash does not mean the pump is OK, the pump gets damaged when water leaks into the well in the trunk and water accumulates.
Remove the trunk carpet and see if water is in the well under which the pump sits, this is the passenger side well. Slide a paper spill down the crack beside the foam blocks in which the pumps sits.
These wells are aptly named as they often have water in them when pump problems arise. Operate the fob with the carpet out and listen.