I have cleaned and rebuilt a number of controllers that have those little asbestos insulators in them. I handle them carefully and then put them back. Don't ingest the stuff and it's a non-issue. Those little insulators are always sandwiched between things. Usually the ceramic resistance unit and the metal outer housing. So using the controller does not cause them to move or degrade.
If they are damaged replacing them with the phenolic washers is a good idea. Now, just where does one get such phenolic washers? I've no idea.
The backs to every cord block I've looked at have either been cardboard or plastic. Never seen one of those made from asbestos.
Joe