Why are large industrial motors three-phase? I almost always see large industry getting three-phase, which I assumed either gave more power or more efficiency.

It has nothing to do with efficiency (do the math on the conductor gauge and it all mostly cancels out) or power (idem) but the types of motors used (squirrel cage I believe).

The problem is getting the motor to start spinning. You cant start all motors on just one or two phases. And if a motor isn't spinning, it's essentially a short.

No, very large single-speed industrial motors are synchronous motors, not squirrel-cage (induction). These motors cannot start spinning on any number of phases: they can only run at the synchronous speed (which is dictated by the power line frequency and the number of poles in the motor). And it's not a short; it just doesn't run at all if you supply power to it; it only works once it's at the synchronous speed. These motors are used because they're the most efficient type of AC motor. Induction motors have "slip" of around 3% IIRC.

Basically what the sibling comment said, it’s easier to start a three-phase motor, and a three-phase motor will also run more smoothly as it is powered by 3 sine waves 120 degrees out of phase with each other which provides constant power; versus a single-phase motor being powered by a single sine wave that outputs no power at the middle of the sine wave with two power peaks, which doesn’t run as smoothly as three-phase power with no points of zero power output.