With respect, I did say in my original post: "I use only the wall switch, not the pull cord on the fan..." but I appreciate your continuing to help me fix the problem, Toymeister.

My lanai ceiling fan failed. On disassembly of the fan I found the epoxied capacitor module (4 or 5 wire connections) with a significant bulge in it. Ordered an exact replacement from Amazon. About $10. Fan working fine on all speeds ever since.

(Healthy as a horse and darned near as smart.)

Since you always leave the pull chain switch in one position, it's almost certainly the wall control that is bad. The pull chain switch is cheap and could easily fail but the fact that other speeds work tells me that it's ok.. Change out the wall controller.

Pulling the fan chain does change speeds - it just never gives me the top speed, which would be the case if the capacitor were faulty for that setting.

Replace the cap
 

The fan won't run without a capacitor. I replace mine with a duplicate from Amazon or ebay; I don't remember. It has been running for years since.

Why not just remove the capacitor and order a replacement from Amazon? Less than 10 dollars and you can get it tomorrow.

Fan Speed
 

FYI: The only variables that control speed of a single-phase AC motor are the frequency and the number of poles in the motor. The frequency is fixed at 60 cycles from the power company. Your pull chain switches in and out the correct number of poles for the speed.

The formula is: S= 120F/N, where N is the number of poles and F is the frequency.
As an example: At 200 RPM the fan has 36 poles. So, you can see why ceiling fans are so large and heavy.

The capacitor has nothing to do with speed. It increases starting and/or running torque while reducing starting current. Pretty amazing, huh!

If the fan will start by hand-spinning the blades, then you have a bad capacitor. If not, then it is the switch (or really bad bearings!) Poles and windings hardly ever go bad.
Good luck! (P.S. I taught motors & controls at Ohio University for 37 years)

Hunter Capacitor
 

Excellent contact info. They replaced mine for free. It's easy to install.

My Uplifting Experience
 

Easy there, fella. I had the same experience, so I installed turbochargers from the hottest Tesla models on each fan, upgraded my electrical system and presented a special new all-ceiling-fans-at-once remote to my bride for her ----th birthday. She pressed the power button and
KA-POW - the entire roof groaned, snapped free from the house and plopped onto the neighbors' new swimming pool, which we are no longer at liberty to use.

Upside? Won special awards for largest electric drone and for thinking outside the breaker box.

Downside? Buying new furniture every time it rains.