This is very confused thinking. I'm not sure where to start here.
Browsers are really the most stable backwards compatible platform you'll find provided you:
1. stay away from proprietary extensions, applets, plugins, and what have you.
2. stay away from the new features that aren't standardised yet.
The different parser models are there, and are STILL there. they will be there forever. Why? because DON'T BREAK THE WEB.
"At best, most browsers these days estimate what they are doing, diving into some wierd mode full of edge cases purely by accident if you step on the wrong stone."
Evidence or it didn't happen.
"As for proprietary extensions, they are the most stable in IE. The IE8 change was the first since IE4 and it was primarily a security model change. "
IE's proprietary extensions are the most stable, except they aren't, at all? Are you reading what you're writing before you send it?
By popular demand IE has been forced to expunge the worst of it, and support standards. The ones that don't change every browser release.
"Now we have NaCl on the horizon (ActiveX v2) and every vendor fighting their own extensions into the "standard" by buddying up for a new "standards" group."
Well then don't use those.
"It's a minefield which throwing critical applications into is a bad move both from a logical and risk perspective."
It's only a minefield if you go into it with extremely misguided and confused thinking like you apparently have. There is a huge, rich, extremely stable platform here that you missed out on because you ironically are only interested in the whiz bang new shiny feeding frenzy going on. Just don't use those parts that are changing rapidly. they are easy to spot. Use the old parts that haven't changed since 2004. If they've been around for 10 years they'll be here for 10 more years.