Some dryers do have motor start capacitors but not of the the kind you referenced in old CRT’s.

That being said, there are a whole host of other safety-related things OP could have been injured by as well when disassembling the unit even if it was de-energized. Sharp edges in the sheet metal panels and drum, pulley arms under tension are a couple great examples.

I would never want anyone to get injured by their appliance because I truly care about everyone’s well being and health, but further, as a former in-home technician I also absolutely dreaded walking into someone’s domicile to find they disassembled their product (usually the wrong way) and couldn’t put it back together, losing screws or damaging components in the process. It creates hellacious headaches for technicians to unfuck the mess a product owner can sometimes cause by uninformed disassembly. Often does not end well, unfortunately.