"..got one of those small, c-type Lee presses that I use for secondary, once-in-a-while work. I never did the whole loading process on it, but one probably could. "
That's Lee's very small "Reloader" press. You can certainly load anything up to .30-06 on it, probably larger stuff too.
I have a couple I got for special tasks (de and recapping). When I got my .001" dial indicator gage I checked lots of things in my shop including the press spring in my RC II when FL sizing .30-06, got almost .003". That's not a lot but it's more than I had expected. Then checked the Reloaders, both, with the same dies and box of fired cases but didn't get enough spring to measure! The small alum alloy presses certainly aren't as strong as my big iron green press but, within their strength, they ARE more rigid and rigidity is good when making quality ammo.
Lee's Classic Cast and Classic Turret are as strong as any other iron presses. Precisely machined too.