Could most AC devices run off of DC rather than AC?
Short answer: No, most AC devices could not run off DC.

In-depth answer: There are very specific cases where this will work. If your input circuit is a diode rectifier followed by capacitors, after which everything in the system is expecting DC, then you should be able to feed DC into the circuit. You should even be able to ignore the polarity of the DC input.

If you have any transformers at all in the AC path, they will saturate, which will at best blow a fuse, and at worst damage something else in the path.

If there's a power factor corrector on the AC input, it's imaginable that it will work off DC, and it's imaginable that it won't. My bet would be that it won't.

Even in the best-case scenario, the losses in the diode pair being used will be a little more than twice the losses if you were running straight AC into them, because only one pair of diodes is all the current that was previously being seen by both pairs. (All this assumes single-phase. Three-phase is comparable, but with a 3x multiplier instead.) So unless the diodes have a large amount of overhead built into them, they're likely to overheat and fail.