Yes, that's how it works.
The IE implementation (if you can call it that) is utterly meaningless from almost every conceivable angle. All they did was to ensure that if you do have a wide gamut monitor, everything will be wrong. No exception. Even if people deliberately posted Adobe RGB for the benefit of the wide gamut users, it would be wrong, because the display gets sRGB.
And of course the obvious point that Simon Garrett made, repeated for emphasis: if you're converting anyway, why not do it to the right profile?
That's why Firefox in mode 1 is so brilliant: aside from honoring embedded profiles, it actively assigns sRGB to all untagged material (including webpage elements) and converts to monitor profile. It doesn't just pass it on to the display unchanged, as Safari (or FF mode 2) does. So you could say that with Firefox, sRGB really is the web standard.