I own a Trek 7700 Hybrid, shocks in front and back. This bike is very good for Rail to Trails which feature large unexpected holes and other obstructions. The lime stone path will get soft during a heavy rain. Not so good for skinny slick tires. So I have 38 mm wheels.
I also got a speed bug. So I have also 700 x 25 mm tires to go like hell on paved roads. With aero bars (same Hybrid) I can average 20 MPH for 50 miles. Not bad for an old guy but not good enough to compete.
I also own a light weight road bike. Cannondale R2000. I am only 5% faster with that bike. It is still a much better choice for hill climbing because of a 10 lb. difference in weight.
Either bike costs more than your budget. I think that this information can give you some ideas. I also started with $600 Hybrids. I can share with you all the problems I had with that. (Schwinn)
MTB is good for its intended purpose. If speed is in your future, they will not do the trick. My son loves his MTB and claims he feels very safe riding it on roads.
He will go on the side strip if he hears a car coming. My son is not an aggressive biker. But he bikes a lot for exercise.