Yanmar YM1510D starts and runs but doesn’t move in any gear

[California]

... The rear wheels appear to be locked, too. ...

Actually that might be a Yanmar feature, and a clue to the problem.

Whenever the tractor isn't moving under power the Powershift engages its clutch packs for two gears at the same time, weakly, to provide a minimal parking brake. The guts inside Powershift are about the same as a small car AT except no torque converter on the flywheel. The clutch shaft goes right into the transmission, first to a hydraulic pump then to clutch packs as chosen by your gear (not range) selector. If the clutch isn't turning that internal pump then it will feel like a parking brake left on.

Now - if the kids were trying to smoke the tires shifting any gear at max rpm, that clutch shaft or the spline where it enters the transmission might be broken.

On my YM186D (same tractor) this braking is very helpful, for example when I get off to water each new tree. No need to engage the real parking brake, at least on somewhat flat ground. The first YM186D I had previous (no loader) had this feature, it was noticeable, but not as strong.

I have to remember the larger YM240 (YM2000) doesn't have this feature and can roll away if I simply dismount. Don't ask ....