I have a similar but less extreme experience. I have a Tommee Tippee prep machine for baby formula milk. The part it has water coming out got so dirty after months of use. Due to lack of sleep like any father of newborn does, I flip it upside down and clean it with running water, and then I found out all the water finds its way into the machine. Then I completely disassemble it, I use a dehumidifier to blow dry air d…
I highly doubt that a dishwasher run would change that.
The real reasons not to use the dishwasher are:
— dishwasher isn't a replacement from scrubbing
— caked-on grease will still be there
— you'd need to clean your dishwasher afterwards (more work)
— it's far easier to shake the bread crumbs out when they're dry (and there's a convenient latch on the bottom for that) than if they're wet and stuck to the insides
— There's a potential for rust to form somewhere inside the toaster, which will not make it work (or look) any better. That's not specific to toasters.
Crumpled aluminum foil does wonders for cleaning metallic surfaces from rust and burned grease though. And chemical solutions like 409 can make everything else sparkling clean.