For what it's worth, road bikes are sturdier than hybrids. If you need something sturdier than a road bike, then we're talking about cyclocross, touring bikes, or (on the hilarious end) downhill mtbs and similar.
That said, I think your best budget bet is to go to bikesnotbombs and find an old road bike that fits well.
Schwinn Varsity are usually old 30lb+ beasts that weren't very good even when new, so don't pay too much for one. If you want wider tires on it, you'll need to be concerned with a) max clearance of the brakes and b) max clearance of the frame where the tire passes. Either way I bet you can put 28mm tires on it, which is the standard "fat road-bike tire". There are wider tires of course that will fit on 700c (think mountain bike-style tires, for cyclocross) but you generally don't want to ride on them unless you're in mud, snow, etc.