(snipped "hand drill press" stuff)
I looked at one of those at Sears and it didn't seem very sturdy. How
accurate/sturdy are they?

I've only tried the HF version, and I was not impressed, but none that
I've seen seem very solid.
My suggestion: drill a ~1/2" hole by hand into your bench top, then
use a plunge router with a 3/4" bit at a fairly low speed and plunge
as deep as it will go. If you need to drill deeper than that, just
use a forstner and you already have a guide hole started. You might
make a larger router sub-base or somehow secure the router so it
doesn't move while you're plunging.
I'd stay away from a hole saw - at 2-1/2" thick, your top might be
thicker than the hole saw is deep, so it might not work at all, and
even if it did, it probably wouldn't leave a very clean hole.
Good luck,
Andy