Yes, it’s possible. PMTCC has a member who rides his 29er on a lot of PMTCC rides. One of the ride is from 16th Bridge or Performance Bicycling, Freeport Rd, New Kensington, 366, 380, back to the beginning.
I haven’t done it on a mountaining bicycle, but I ride from Monroeville (near the interchange with the turnpike and William Penn Hwy) to the east end daily. I’m typically on my steel touring bike (curb weight ~32 lbs) with about 8~12 lbs of totally necessary gear crap.
FWIW, the drivetrain on the tourer is a deore MTB setup. I suppose the only big differences between my ride and a typical hardtail might be fork, handlebars, and geometry; weight and gearing should be about the same.
I think the easiest way to Forbes Hospital is Penn to Greensburg Pike, down Greensburg Pike to Turtle Creek, follow I guess Airbrake to Wilmerding, then the Triborough Expy/Broadway to Pitcairn. Then up Mosside/48 to Forbes. There’s no shoulder on Mosside after Macbeth so you could take the back streets to the left to get to the top.
I would recommend Frankstown Rd to Jefferson Rd**, then follow Old William Penn and cut over towards the Miracle Mile wherever you’re most comfortable. Personally, I would take OWP all the way to Haymaker, go right and follow that to the hospital.
I’ve done this route in the inbound direction after work, but never outbound. It might be a completely different ride depending on morning vs evening traffic and your direction.
Edit: I’ve taken OWP outbound during the morning rush hour, but never Frankstown Rd nor Jefferson. It wasn’t really that bad traffic-wise, despite their being limited shoulder space through certain areas.
**Jefferson to OWP should be an AWESOME descent, but I’ve only ever climbed it. Watch out for the STOP sign at the bottom of the hill!