Try installing traditional incandescent bulbs as a test. Use 40 or 60 watt bulbs as most light kits are rated for a max of 60 watts each.
If the limiter is a 2 wire device, the limiter will need to be temporarily removed and the 2 light wires temporarily spliced together in order to by-pass the limiter.
Yes, I understand your concerns on testing this. Since the light kit contains a plug-in harness connection, locating the hot and neutral point for the light kit requires further evaluation and testing.
The quickest and 1st test method to isolate and troubleshoot the problem is to take voltage measurements directly inside the light bulb sockets.
It is possible that the wattage limiter has become faulty. An exact replacement type would be required. Unfortunately, Home Depot or Lowe's does not sell wattage limiters. Yes, they can become defective over the years.
Another possibility is that there is a loose wire connection inside the wire harness or pull chain switch.
The problem won't be the capacitor, those are only used to control the speed of the fan motor.