I'm personally using Firefox, and I never really saw it go above 200mb in usage (I'm a light surfer, that restarts his browser every few minutes and only keep three tabs at max open). When you combine that with the amount of ram that the alpha of BF3 used (which was the highest of 1998mb in my tests), you only get ~2.3gigabytes of ram used. These days, the minimum for people that play PC games, is four gigabytes. Now if you'll try to run BF3 with only 2 gigabytes of ram in total in your system, not even closing the browser and launching Steam instead of it going to help you gain more performance out of the game.

And Steam does force you to run Steam in the background and be logged in to your account (in offline mode or not). Origin on the other hand, does not. After you downloaded the game through Origin and it installed it, you can safely close Origin and launch your game directly without it. Here's a fine example of it in action: Launching a singleplayer/multiplayer game that was installed through Origin: It launches without opening Origin after you input the game's cd-key. Launching a singleplayer game that was installed through Steam: It launches Steam instead, for you to log in to your account
And it's well known that Steams off-line mode, isn't always properly working. Even sometimes when you'll unplug the ethernet cable/wifi card; which is the most known fix for that problem.

Now I don't know where you've read that BF3 will require constant internet connection to play the singleplayer (I'm not arguing for that fact for the MP part :) ), but I'm gonna call bullshit on that rumor (if it's even a rumor). Even in the MP only alpha, the game had signs of the singleplayer DRM (if you still have the alpha installed, check the "core" directory). It was the same DRM as in BC2. Which the only authorization that required internet there, was the MP section. And it was activated using your EA account, just like battlelog.
The singeplayer section was separate and only required cd-key authentication, and if you didn't have Internet connection, could be done using the DVD check.

And until the game going to be released or someone from DICE/EA going to say otherwise about the DRM on the singleplayer; that's the version I'm going to stick to, as it's the only one with some "back" behind it.