Lots of shops offer to cover your deductible, and it's not shady.
It would be shady if they covered your deductible and then tried to bill it to the insurance company, but that's not your problem if they do. And all the insurance companies know that trick, so they won't try.
Here's the bottom line with your deductible: It's the part of your loss that the insurer doesn't pay. They don't care who does pay it, as long as they don't. If the body shop, or your mom, or a total stranger wants to cover your deductible, the insurance company couldn't care less. That's why they said don't bother asking the insurer.
So if they put "deductible waived" on the final work order, it means you don't have to pay it. And if the insurer doesn't approve of that, they'll take it up with the repair shop. You're off the hook either way.