"What would be a good camera to get started into the world of backpacking photography?"
Hooboy, that's a wide open quesion. You're right that a lot of P&S cameras either don't allow control overrides or they do, but the controls are buried deep within demented menu systems. Not ideal for shooting on the go!
Any camera with a "PASM" mode dial gets you to the desired settings, so at least insist on a camera with a dial control. If you want to stay small, several pro-quality compacts from Canon, Panasonic, Ricoh and the like will fill the bill, and add RAW file capability for the time you want to dig into extensive image manipulation on your computer.
Most "superzooms" have mode dials and of course all dslrs do too, as do the relatively new mirrorless system cameras from Olympus, Panny, Sony and Samsung. These, of course are bigger and heavier than P&Ss, but you may find yourself migrating towards them anyway, due to the vast lens selections.
Used cameras are a nice option for anybody on a budget. I'd guess you could get, say, a Canon G9 for a relative song. It's a solid and fully featured compact from a popular series, with very good controls.
I'm hesitant to recommend specific new models without know what your size, weight and cost targets might be. As you know, there are a thousand models for sale on any given day.