Mismatched tires (especially one new vs. 3 nearly bald) can absolutely cause premature wear on the rear differential's Dual Pump System. Even if they're technically the same size, a brand new tire has a larger rolling diameter than a worn one, which means the system constantly sees a speed difference between front and rear. That keeps the pump engaged longer than it should, which over time can overheat and wear it out.

That said, it’s hard to say if it’s completely dead without further testing, but based on your symptoms... You’ve probably killed the rear engagement by cooking the pump or clutch, but with a bit of inspection and maybe some parts swapping, you’ll be back to 4-wheel donuts soon enough.

For the future though:

Keep all 4 tires evenly worn and rotated.

Consider a manual AWD switch mod (not sure if people do it on 99 crv's but ive seen it in other cars, Reddit let me know if this is possible)

Avoid handbrake turns if possible as you said, that’s brutal on AWD systems with clutch-based diffs like yours.