The reason they're flash based and run in a browser is because that way you don't have to develop a client side application. Anyone, from anywhere, with any kind of flash compatible browser can play. It may be inefficient, but it's universal.

For the more sophisticated games, you need to create a client application. Users have to trust you enough to download and run your client software. Given what's on the net these days, that can be a hard sell (who's to say your client isn't another spyware app?). Then you have to support the client, running on who knows what hardware, with who knows what version of operating system, etc, etc. Some users will be excluded from your game by way of what computer they use (Mac, Windows, Linux, other). All in all alot of extra work that takes you away from your real goal of making the best darn game server you can.

Communicating over the net doesn't have to be http. You can use any port and what ever communication scheme you want over a bog standard TCP connection. The client doesn't have to be a browser, the server doesn't have to be a web server, and neither of them have to follow a valid HTTP protocal. If you're not running the client from a browser, then there's no reason to use HTTP, but it does have one particular advantage in that it usually won't be blocked by firewalls (the port that is).