score:6
This info may help others ...
I took a huge risk and simply applied again.
Very fortunately, it worked.
So, within seconds of clicking "pay", I got the actual eTA via email, including the eTA number.
For anyone facing this total catastrophe, note that
Interestingly, this means that if there was some problem I can not interact with them in any way. You must have the "application number" to lodge any sort of communication.
(Indeed, I will never be able to get back the $7 from the first attempt!!)
Again to be clear, regarding my first attempt it just utterly failed in every way possible. The $7 charge went through correctly, but (i) I did not get the "you've applied" email, hence (ii) I have absolutely no way to interact with them, they demand that number and (iii) I did not get the eTA
Regarding the second attempt the email which staggered in to my email box was highly "broken", all the headers were f'd up, the formatting and grammar were comic, etc.
Surveying the internet, this is clearly a crisis-level problem that is plaguing many folks trying to "go to Canada".
The idea of applying again is a nightmare, you can see it could drastically gum up the works. Fortunately in my case it did work as described.