"First Home" (mfgr) 3 speed ceiling fan went to 'high speed' only last night.
Pull chain only clicks on at 'high speed' and merely clicks noise at lower speed positions.
It takes 4 clicks of pull chain to stop and restart at high spped.
Reverse switch works fan in reverse spin also only at highest speed.
Is the pull chain mechanism the most likely problem?

Could be a wire out of fan switch, a bad switch, or capacitor(s).
Visual inspection can sometime reveal(Deformed capacacitor(s),loose wire, melted switch).

MOST LIKELY capacitors. Very rarely is the switch the problem

Ceiling fan capactors - troubleshooting and resources - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-capacitors.php)

Shopping to find stores that have replacement capacitors is as wild goose as it gets.
I went to Home Depot then to Colonial Electric, then to Rumsey Electric, then to K A Y Electric and NO LUCK!
Went back on line and found
Ceilingfanparts.com and ordered appropriate capacitor for about $12 plus $8 shipping.
Expect that the knowledgeable woman was the best deal of the day. Part is now on the way and I will come back after replacing zapped capacitor to restore all speeds to give an update.

That's the site I would have sent you to. I hope this solves your problem.

Additional Question re: installing replacement capacitor with its wire leads:


Will I have exceptional difficulty after adding about six wire nuts to reconnect black & white power lines plus the four capacitor wires? OR
Is there an easy way to remove the four old capacitor wire leads from the pull switch et al (reverse switch & power harness/clip) without destroying the switch itself?

Yes, the leads come right out of the pull switch. Follow the instructions on the pull switch replacement page (except in this case you are replacing the capacitor, not the pull switch)

Ceiling fan pull chain replacement and repair - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-pullchain-replacement-and-repair.php)

Should've referred there sooner 2 avoid wild goose chase amongst local shops.

OR you could have googled 'ceiling fan capacitors' I believe it is the fourth page that comes up-- the first being the replacement instructions I already posted, and the second and third being two other sites that sell capacitors. Do people expect you to do everything for them?