I ride an early '90s rigid mountain bike and commute five miles each way. It's probably 25% dirt/fine gravel and 75% pavement and it's hilly.
I use Forte Gotham street tires available inexpensively at Performance Bike. I think I paid $11 each on sale in a store. I live in goathead burr country and I haven't had a flat yet, although they're still pretty new.
For me, the much easier/faster riding on paved roads far, FAR outweighs any weakness off-road. I'm not riding technical single-track but just trails and fire roads and these Gothams are perfectly fine for that.
The only off-road slippage I've noticed is when riding from dirt up onto a cement or pavement - with knobbies, you can approach at just about any angle to hop up on the pavement and the tires will grip fine. With these, a more perpendicular approach to a paved surface that's a higher than the little the dirt is safer.
I've found if you try to hit that pavement at a very low angle, like say 10 to 30 degrees (I'm just guessing here), the sides of the tires can slip against the pavement and could put you on the ground if you're really unaware or going too fast to recover. But this is rare - I did it once, recovered without wiping out, and now I just watch out for that situation.
Forté Gotham Road and Mountain Tires