That would be cool, I had the same idea, but you actually have the skill to make it happen! (in fullscreen
So I'm thinking Squid. Caching web proxy. Could probably even write some extensions to it to make websites Atari friendly, then You'd have everything redirected to the browser on the ST. Pretty sure that wouldn't be too difficult (stripping out all the crap might be harder, but not impossible)
I may have a play around and see if that'll work.
Atari 8Bits: 800xl, 600xl, XEGS, 800, 130xe, 130xe (fully upgraded (soon!))
Atari STs: 1040STf (broken shifter), 1040STe, Mega STe, TT030, Falcon (CT60e, SuperVidel)