I can't tell definitively from the pictures - but it doesn't look like simple overload failure. Simple overload causes teeth to be broken off. Insufficient lubrication will cause the teeth to wear until they are too thin and the remains get sheared off.
This sort of wear (due to poor lubrication) was the cause of failure of a drive nut in the Alaska Air crash a few years ago. See
figure 20. There was no warning to the flight crew though backlash checks (which were skipped) would have certainly detected the problem.
Since it is continuously driven, it's possible the typical load is too high for any lubricant to matter, which would cause accelerated wear. This might cause fatigue to damage the the worm face, leading to a high-abrasion rate. Is the worm in good shape( smooth) or is the worm tooth face rough? If it went 15 years, that's not likely.
For the service it is getting, I'd suggest oil rather than grease. In cases where there is a wiping action, grease can be ineffective. Worm drives are all wiping action connections. If the oil-film can't reform fast enough, it's like not lubricating it at all. Also, I agree with what Tmoose said.
For a quick check, get a power meter on the motor to see that there isn't a periodic high load from the conveyor. It is certainly tempting to think something like a bent shaft causes a large increase in load that hits the same segment of the worm gear each turn, causing just that segment to wear faster than the rest. Motor inertia would push through the bind and have most of the turn to get back to speed. There wouldn't be a noteworthy increase in noise.
I agree on the PM plan; make a measurement of the gearbox backlash by first driving the input to take all the clearance, then locking the input and turning the output in the same direction. This should avoid including slop in worm support bearings in the backlash, and will also make it repeatable for later. Repeated at intervals it should show a linear increase in backlash as the worm and worm gear wear. The rate of wear will increase dramatically if surface roughness increases or lubrication fails. The trick is figuring out a backlash limit. Making a measurement on the existing box (with the remaining teeth) could provide a guide.
For the longer term, especially if using an oil bath, monitoring the power into the motor and the temperature of the oil vs ambient will also indicate a change in operating condition. The value of this is adding predictability to the down time.
Alternatively, put in a P.O. for a new gearbox, delivery date in 14 years, 6 months.