I had my 1983 Series 3 refurbished about 5 years ago @ Sota, and Kirk (R.I.P.) said my motor was in great shape. He was amazed that it had been a demo table for about a year before I bought it and even offered me a sizable credit if I was ever interested in upgrading to the new motor with the internal power supply. So while anything is possible, the motor may not be the problem.
The outboard power supply itself and the AC power should be checked. The old supply with the shaped brown plastic may no longer be meeting specifications and they are known to fail. In my case, the transformer came loose in transit and shorted when I plugged it in after getting it back (it was hot-melt glued in position back when). I decided to buy the new replacement in the much better looking black rectangular box instead of fixing the old one. Next, if you have lower than normal AC voltage (a problem in some older grid locations), the PS may be under supplied. Both the AC and DC can be measured with a simple multimeter.
I've never had my belt cause a problem by itself; I ran the OE belt since I got the table and it was always rock-steady on spin-up and speed. Ditto for the new replacement in 2012. FWIW, my table takes about 30 seconds to reach stable speed since the new bearing, platter and suspension were installed; slightly faster than before the work was done.
You may also want to check both the bearing transit lock grub screws and the motor height adjustment screw (accessible through the sub-chassis directly under the motor). The grub screws can drag on the bearing and should be backed out from the bottom of the sub-chassis at least 1/4 inch. I have mine backed out about 1 inch; I have the room and they're very long.
The motor height can affect speed, belt wear and/or create noise. It's usually correctly adjusted when the belt is centered on the convex pulley. This can be seen by removing the speed control cover. If it's off to one side or the other, you can use a thin flat-blade screwdriver to adjust the screw slightly. This is to be done while the motor is running. The Sota documentation says the motor should be run for 1 to 2 hours afterward to verify the adjustment is correct.
Good luck and let us know how you make out!