Lighting – Could switching from a dimmable to a non dimmable LED bulb have caused a light fixture with dimmer switch to fail

Yes, using non-dimmable lamps on a dimmer will eventually damage one or the other. Usually the lamps will suffer, specifically the driver in the LED.

Using the dimmer on full high should be OK, but you have to be careful to keep it that way.

Dimming LEDs

For a typical ceiling fan, the light fixture is exactly that-- a light fixture. It is nothing more than wires, a chain switch, and bulb socket(s).

Dimming LED bulbs in a typical ceiling fan fixture will be completely dependent on ensuring a few things:

1. The LED bulbs are dimmable
2. The dimmer switch is rated for use with dimmable LED bulbs
3. The particular models of dimmable LED and LED dimmer play nice together (this doesn't always happen)

If you have a fancier ceiling fan with integrated remote features, fade on/fade off lights, etc., the actual fan fixture could come into play. However, from your question it does not sound like this is your case.

Illuminated Switch with LED load

First, see here: How does an illuminated switch light up without a neutral

You have a few options:

1. Look for an illuminated switch designed for use with LEDs (would need a neutral wire). I've never looked for these before so I can't say for a fact they even exist in production.
2. Buy whatever illuminated switch you want and hunt for LED bulbs that will just so happen to operate correctly with the switch you buy.
3. Buy whatever illuminated switch and whatever LED bulbs you want and use a single incandescent bulb in your fan. The remaining bulbs can be LED.

As you've probably noticed, there could be a bit of trial and error in this process. LED bulbs are not like incandescent bulbs where all bulbs were virtually the same and therefore behaved the same. LEDs are driven by electronics within the bulb housing, and these electronics can and will vary from manufacturer to manufacturer, and even from product to product within the same manufacturer.