HLTV would be great
FFSplit for Windows uses ffmpeg (which you claim to be garbage)
I've linked to 2 sites with guides for what you are requesting.
https://gist.github.com/3178130
http://www.creativetux.com/2012/11/streaming-to-twitchtv-with-linux.html
It took me less than 10 minutes to get it working as intended.
It is not because I disagree with your request, while it would be 'neat' that Steam would do it for you, remember it will just get more dependencies.
Read carefully, I didn't say that ffmpeg is garbage, but that it's not easy to setup for complete Linux newbie. We need simple, easy to understand GUI software for broadcasting.
May I suggest you put Steam feature requests here http://steamcommunity.com/groups/SteamClientBeta :)
Nothing extracted yet.
Windows users have tools like Xsplit or even EA Origin if they want to broadcast their gameplay to services like twitch.tv or just use Fraps for local recording. There are some solutions under Linux, but most require some effort to get them working (ffmpeg, glc) or are complete gargabe when it comes to 3D content (recordmydesktop, kazam).
Built-in screencasting tool could help promote Linux gaming and Steam itself.