How can I make Caps Lock work like it did prior to Sierra?
I think I may have found the solution:

It appears that in Sierra, the "default" behavior that I described above is possible, and is now "a feature". Only problem is it's turned-off by default (which is plain ludicrous). 

There is now a new option in the input sources screen, which, when turned on, reverts the behavior to the desired one. 



The Caps Lock key longer turns green when non-Latin languages are used, but the keys behave as expected. There's the annoying symptom of accidental switching between languages for those who use Caps Lock to Capitalize in English. Luckily for me at least, I switched to Shift for that early on.