It can be done with an old steel frame and some decent aluminum rims.
I had kind of the same idea as you, but I wanted a few more gears. I took my mid 80s Trek road bike (old 12 speed) and had my LBS re-build the rear wheel with a 7 speed Nexus hub. I removed the derailleurs and moved the 42 tooth chainring to the spot previously occupied by the 52 tooth ring (yeah, I'm a weakling, but at least I'm comfortable!). I initially was going for the full-on upright posture of a 3 speed, but I've since realized that I like the more efficient posture of the drop bars and that my frame geometry is a little steeper than those slack angles of the classic 3 speeds.
The biggest improvement that I've made over the old 3 speeds is the higher quality and lighter frame and the lighter aluminum wheels.
The 7 speed hub is a little tighter fit between the chainstays, but it works. The back of the bike feels a little heavier than it used to when I lift it, but riding it's fine. Best of all, no missed shifts, no downshifting at lights.