What I would do:

• Get a 90s steel mountain bike with 26" wheels, a rigid fork, and rim brakes.
• Do a drivetrain conversion that gives you a 1 X 8 or 9, with a 32 or 34T chainring and an 11-32 cassette.
• Keep it moto and use a flat or riser bar. If you want to run drops, spring for a VO Cigne stem; you'll need the height.
• If the brakes are cantilever, switch to V-brakes and appropriate levers.
• Continental Double Fighter 26 X 1.9" tires, with tubes.

It's probably not shoestring budget-friendly, but you would be well equipped for a lot of different conditions and surfaces without any further futzing around. I don't know how much of this you'd want to tackle yourself (some shop tools required), but it's not an overwhelming endeavor, especially if you've got Co-Op or LBS help on standby.