Ceiling Fan capacitor question

If you install a remote control module, it will control the speed of the fan. Which means, there will be no need for the pull switch. So yes, you can install a remote module instead of fixing the medium speed.

Install the remote module, put the pull switch in high, and enjoy.

While the observed behavior does not correspond with your conjectured wiring description -- it does correspond with your suggestion that the fan and light switches were interchanged. Based on your diagram, the following wiring should work:

1. Black from house nuts with White to the light switch (tag this wire with a piece of electrical tape!) and one of the Blacks on the fan switch.
2. White from house nuts with White to fan.
3. Other Black on fan switch nuts with Black to fan.
4. Black to light switch nuts with Blue to fan.
5. And of course, all the grounds together.

Last but not least -- I suspect the Blue (probably actually red, but w/e) wire to the fan and the White wire to the fan were swapped at the fan from your description -- the behavior in your description would have the fan running all the time with both switches controlling the light. You'll want to fix that while you're changing the wiring in the fan switch box -- the best way to do so is to go into the fan box and switch the blue and white wires if white isn't to white and red or whatever isn't hooked up to blue.