How does a speaker blow out?

You got 22 watts, you can't play loud on most loudspeakers with 22 watts. Low power is bad for loudspeakers because the waveform comes to a sudden stop and the driver goes out of control. Damage happens long before you hear the distortion. Turn it down and live within your system's capabilities.

Well, the theory is actually that the clipping causes a square wave, which has a lot of high frequency harmonics, not DC exactly, so more likely to blow a tweeter.

It can happen for a lot of reasons including age and prior abuse.  There are two kind of failures.  Mechanical and electrical.

Mechanical means that the surround or the spider has failed, either from age or excess physical motion.  The electrical failures are caused by the melting of wires in the voice coil or to/from it or a disconnect.

It was a woofer, so the low power seems to be the issue.  The amp is triode/ultralinear switchable and I liked the triode mode better.  I should have switched to ultralinear (i.e., 45 W) if I wanted to play loud, it seems.

The difference between triode and ultralinear is not going to be enuf, I think, to protect your speakers when your blasting away. Depending on your stuff the difference might be only about 3db. Save your speakers, your ears, and your neighbor's good will - keep the volume down. :-)

       What Erik said (regarding clipping), but: after a x-over passes that high freq distortion to the tweeter and burns it (typically: open), the energy that no longer has a path that way, is routed to the next highest freq driver, in the circuit.

                  In the case of a two-way: that would be the woofer.

        I lost count of the number of two-way systems in which I found both drivers toasted by an under-powered amp, while in the reconing biz.    Blown x-over caps, as well.

        Though a clipped signal isn't DC; it certainly can do as much damage.

         On the other hand: hearing a, "...most painful noise, coming from one of the woofers", might indicate a woofer that was driven past it's mechanical excursion limit (X_{mech, per Thiele-Small)}, bottoming the voice coil, or: causing it to hit the gap's top edge(s).

         That type of damage usually resulted from boosted/heavy Bass and a vented cabinet, regardless of amplifier rating.