I don't think Dart will beg other browsers to comply to their will of adding a VM.
On the other hand, I find it more likely that, in a whole year's time, the people behind Dart thought about NPAPI. They may very well produce a plugin that everyone will download.
That would make Dart the new Flash, but that's the price to pay for a serious cross-browser Dart.
Of course, that would also make Dart second-class citizen in the web platform, and I guess Dart people don't want that. However, I find it hard to believe that they hoped all browser vendors would bow before the new programming language overlord.
Now, making browsers speak Dart will be the easy part, I think.
The Enterprise will have a hard time accepting a brand-new language blindly when they have just learned that C# is slowly losing Microsoft backing.
Programming languages all make mistakes, and Dart will be no exception; they had a full year and bright engineers, but that rule is sacred. No matter how hard they advocate the niceties of the language (which I assume will be closures (again!), the full funarg, yet another garbage collector, optional typing, type inference, bignums, modules, event-oriented programming, maybe lazy evaluation) but I know many programmers that won't bother learning any programming language that falls too far from the C-like tree, especially if it has odd features, especially if there are mistakes in the way those features work together.
I have seen many bright programming languages that never found any serious adoption. Scheme would have been very interesting in real work environments; Perl6 has many interesting features, but is too late and slow; Haskell still has potential, but I don't see it suddenly go past Objective-C, C# or Java in popularity.
I would love Dart to find success. But this is not a dream world. The platform it seeks to conquer, the competition it must overcome, the tender age it has: Dart is so very late for such a challenge.
This has been studied, I don't have the stats but it may have been in the context of Gears: users don't install new plugins lightly. Distribution deals could be bought and paid for, but that is a long road (see ChromeFrame).
Also, NPAPI is not expressive enough to enable a new programming language VM to be integrated on par with JS. Big deal, Google can write browser/OS-specific code? They tried with Gears. It's hard, and versionitis/unfrozen APIs bite hard.
Take these two points together and the NPAPI route is not going to get Dart widespread adoption outside of Chrome, any time soon. So would web developers use it if they had to compile to JS for too many visitors?
I doubt it. CoffeeScript is a transpiler and it may actually speed up code compared to writing in JS. It is popular, with slick RoR integration, but it's not taking over the world from JS.
Dart's new semantics (at least new number types, per the memo) will result in slower compiled-to-JS code, although the win may be programmer productivity. It depends on how much slower, but the first barrier with compiling is getting devs to put up with the toolchain pain.