I have a Roper RAX7245AW washer that was purchased in 1994.

It agitates normally, but when it comes to the end of the agitation cycle, it stops. The motor does not run, it doesn't pump the water out and it does not spin.

I checked the door switch and it works fine.
?
How should I go about diagnosing and fixing the problem

I don't have an answer for you. But, by my posting a response to your question, I am hoping that one of the several fine knowledgeable in appliance repair people on this site takes notice of your question.

I have a similar problem. But my swtich was faulty; I unscrewed it and part of the molded place for the screw was cracked and the unit was cracked open at the glued(?) seam. I clipped the wires and spliced them together (by-passing the lid safety - I know... but my plug is a hand reach away on the shelf behind in the strip bar). Also had checked the drain hose and she is fine (from back of unit the water in drum came spilling out... so that's not clogged up).

The unit does aggitate, but stops at spin and/or drain while the timer(?) ticks...
I'm now at a loss as what that is or what to do.

Opps... PS. Mine is also a Roper came with the place bought in 2005 and the folks who bought it new were here for 8 years or so. Guessing roughly close in years to original poster's machine.