I would argue that GWT is an example of how it can be done, and it's very hard to beat a well-written GWT app (for the client-server part it's bloody hard to beat a badly written GWT app too).
The problem is that those ancient web api's were effectively javascript windows.forms emulators running on the client getting their calls from the server side. That, of course, sucks, because of the latency. Stuff that actually pulls your program apart into server and client pieces is beautiful.
The problem, of course, is that doing it the right way requires rewriting a compiler from scratch. Doing the "emulate the UI library and hang it behind an RPC" is easy, but sucks.
You get what you pay for.
It is possible, somewhat, but you get very ugly web api's, because they are 'implementation before contract' instead of designing an api first, and then letting both sides adhere to it.