When building web-apps, we write code for the server side and for the client side. On the server side we can use any language we want - just give API (REST as standard nowadays) and be happy - any app, written in any language (with access to network), can communicate with your programm.
TypeScript and even Dart, at the moment I write this, should be transpiled into JS to run in all major browsers. On the web page we can use different JS libraries, jQuery plugins and each one will do it's task and they will not do something in cooperation - in the best case functionality of one library (often jQuery) will be used as a tool in other JS libraries. It doesn't mean they can't coexist - they can and do, but often by the price of tight coupling and only in the scope of one language, so even if you transpile Dart to JS, you can't use all the power of Dart just because of compatibility. And it's the reason why there is not so many new languages for the client side.
IdeaI think we could use SOA inside web page, when web app is the set of services, communicating by APIs. So we need transport, like HTTP, and as this transport we can use DOM (what most JS libs use now) or Local Storage (SessionStorage). DOM (and especially Shadow DOM) looks more attractive, because here we can use object.observe and some other interesting things, but DOM requires browser's attention and I'm not sure if this approach is safe in terms of occasional re-renderings, leaked nodes and collisions/pollution with user-defined elements.
In Session Storage we can have more freedom and it's just less dangerous. If we can do dirty-checking with speed of 300 000 checks/ms in JS, then maybe we can something similar with SessionStorage (or DOM at least), and then we can build event dispatchers on top of that and use it as transport for APIs interoperation. It will not be possible to send pointers of functions (as we used to in JS), but it's how a lot of amazing apps work on the servers and on desktops.
If it's implemented already somewhere - please let me know.