> HTTP components in language API's
I'm going to assume you mean complete stack (HTTP, HTML renderer, JS Engine etc). While the HTTP layer is fairly simple (and thus implement well in a ton of APIs for each of the major platforms), the rest is BIG and to do it well is hard so it is not done well all that often. So what you end up with is either a re-skinned IE, Firefox or Chrome.
Now these exists, for a bunch of different reasons for example before IE had tabs a tabbed IE existed and people who want to stay in the late 1990s have SeaMonkey. However most end up with an interface that is a lot like Chrome/Firefox/IE (as it is a good interface) but without the plugin support.
> Though I wonder how many companies would block you from accessing their site if the browser doesn't have the correct branding
I use to happen a lot and that is why we now have stupid user agent strings.
Mine is currently:
Mozilla/5.0 (Windows NT 6.3; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/33.0.1750.117 Safari/537.36
Mozilla/5.0 - Pretend to be Firefox
AppleWebKit/537.36 - And Safari
KHTML - Base of Safari/Chrome
Gecko - Back to Firefox
Chrome - I could be Chrome
Safari - Of Safari
The only one missing is IE. And I've seen many a user agent string that includes that as well (while not actually being IE).