I wouldn't try the drill press. It is easier to carry a drill to a post than a post to a drill.

For that many mortises have you considered using a t-auger? If it is sharp, it cuts well and the slower pace allows good control.

Aside from that, I would get a 1/2" drill with a big long handle out the side and a clutch to prevent injury and a 2" auger bit and make a guide to show you square in 2 directions and drill them that way. (I simply screw 2 1x6's about 7" long together in an L shape making sure the ends are cut square and then place it where the bit is in the corner as I drill the hole--it's a cheap, quick drill guide).

Or you could get a boring machine like the one in Jim's photo. They work well.

The self-feed bits such as plumbers/electricians use are practically worthless for drilling mortises because as daiku said they don't clear chips.

Also, I wouldn't worry about the bit wandering when drilling to typical mortise depth. Your chisel can make quick work of correcting any slight misalignment. I would drill any through mortises from both sides, though.

I wish I had a good source for a 2" bit, but I cut the end off a T-auger to get mine.

Good luck,

Gabel