So the answer to “Can you play through Bioshock using only The Wrench as a weapon?” is a big “Yes”, at least in Easy mode. In fact, since you’ll actually be equipping the plasmids like “Gain health and eve from hitting people with wrench” and “Extra fast move and hit people with wrench!” it’s actually -easier- to finish with the wrench than when you’re firing shotgun rounds in wild, uncontrolled bursts.
I definitely need a new computer, however – my poor old machine can barely render Bioshock at any kind of frame rate unless I’m playing in 800×600 or .. MAYBE.. 1024 x 768, and if the latter I’d better have the textures set to “Muddy Looking Shit”.