For me, I only use the browser when I'm not around a laptop or desktop machine... otherwise i refrain from doing anything on it.
If I need directions, I use MAPS. If I need webmail I use the GMAIL interface. If I need Chat... well I use Googletalk and other SMS systems.
I use my smartphone as a backup if I don't have ready access to anything else.
Your SIII as a baseline is setting the baseline *VERY* high and either you get a better looking/larger phone or you get disappointed by any other lesser phone.
As for websites almost fitting... Its why you zoom. Many/Most don't understand dynamic width designs. Even Google struggles with it. Heck Apple struggles horrendously with it.
Trying to enter comments here, I can zoom the page but the text box for the comment is fixed to the column count defined on the page, at the text size set on the phone. This textarea happens to render wider than the phone, and it doesn't side-scroll well.