I'll admit I always have the opposite problem -- opens but is slow to close.
Lets see if we can puzzle it out.
Switch failure --
When you activate the switch does it sound like it is getting power?
Have you tried the master switch on the driver's door vs the switch for the specific door?
Can you remove the window, then activate the switch? (not sure if there is a failsafe on it that prevents that - but worth a shot).
From everything you've posted, it doesn't seem like it is a switch failure.
I want you to feel comfortable that the switch works and moves the important players as needed when not under load (no window).
Mechanical --
Swap the motor from a door that does properly lift the window. Does the window raise?
If so, then your original motor is failing or has failed.
Electrical --
Totally out of my element in how to check if the power is wonky. Did you try Aspen500's recommendations?
My problem with windows has always been won't raise issues: (87 5th ave) motor was weak or motor needed tlc on internals, (97 concorde) accuater in door failed, (97 ford) nephew stuck a stick in the window channel.
Not sure this was a very helpful post, but I do feel your pain.