The behaviour you describe is as designed: stereotypes are not inherited but are copied from parent to child on applying a generalization connector. Remove the generalization and the stereotypes stay copied: if they'd been inherited they would've been removed. You can use EA's behaviour to choose whether you want to apply stereotypes singly or all down the hierarchy, simply by changing the order you apply stereotype and generalization.
Surely you jest!?! 
How can I, over time, have 'a priori' decidability over how the model will develop? The model develops as the model will - as a result of analysis and design! The rules around copying of stereotypes have to be declarative, not procedural!
Once again, I'm no longer the modeller, EA is... IT, NOT ME, decides when it will copy the stereotype, based upon some serendipitous set of circumstances!

At a minimum, EA should ask whether it wants me to copy the stereotype. Both at the time the generalization link is created and if I change the stereotype of the Generalized class and a Generalization link exists! (Recursively please!

)
In my view, however, the fact that EA
does copy the stereotype by default on the inital generalization is yet another indication supporting the generalized notion of inheritability of stereotypes!

That's not good: it works for me. Is your metaclass stereotype in a profile or did you define it in "Configuration | UML | Stereotypes"? What is the file format of the metaclass file? What is the name you've given the stereotype?
I don't know where it came from. In any case, from your foregoing, the name of the stereotype should not be relevant! You said, it will copy the stereotype on initial generalization (giving me the choice - see highlight in read above). Certainly, if I do it with one of my own stereotypes - added as you describe, it doesn't work!
Paolo
Consistency, Consistency, Consistency!