The idea of compiling to JavaScript has been done a lot. Microsoft Labs had a project to take CLR codes and compile down to JavaScript. It was abandoned as too slow. I'm sure improvements to the JavaScript engines have made it more feasible. But, as noted the lack of certain native types and immutable data types (i.e. DateTime) forces a ton of static analysis to be done just to figure out that hey, that variable could be an plain integer. And it has to be conservative. Much easier to just have integers and be done with it.
If there is such an insistence on making the web an application platform for everything, then I think at some point you just have to standardize on a VM. Yes, yet another one. So, you can use Dart, JavaScript, Scheme, C#, Java, whatever there's a compiler for. Treat the DOM as core API and enjoy.
Personally, I just hope people realize that operating systems have been doing this well for years, that sandboxing isn't unique to web browsers and that "native" applications are actually a good thing.
The mobile app thing gives me a bit of hope, but it's sad that people seem unwilling to download a installer from the web from a trusted source and install it. I find it a bit strange that people are turning to the web to solve a problem that the web greatly expanded the widespread propagation of viruses and other malware.
And what people are surprised by is a bit baffling as well. A web browser isn't magic. Being impressed that you can run Doom on a modern web browser is missing the forest for a tree. I could run Doom on my computer for ages now. Me having to visit a URL to do so isn't not a major actual change nor improvement, despite the technical accomplishments that went into it.