To me, the sweet spot for what you are looking for is mid-late 80's stuff. Pretty much every major manufacturer here and abroad made a model or range of models to do exactly what you want.

I'm a big fan of late 80's Shimano products, as they work well, index properly, and last a long, long time. You don't specify whether you are going to use downtube shifters or STi/Ergo brake/shifter levers, but the Shimano stuff is pretty much dead reliable riight through today. Some folks prefer Campagnolo, but personally, once they moved beyond friction 6 speed I lost interest.

So...
1987-1988 Schwinn Tempo, Super Sport, Peloton, Prologue, Circuit, Paramount
1986-1989 Centurion Ironman
Most any 1983-1989 Ciocc, Tommassini, Merckx, DeRosa, Bianchi, Colnago or similar well known overseas brand
1987-90 Cannondal SR
Any road Trek or Specialized from that period.

There are plenty more possibilities out there, and channces are you will be able to buy/build exactly what you want within your stated budget, although finding frames that will easily fit tires larger than 25 will be a challenge.