With older drives, pre-about-2010 I think, you can, as I understand it.
After that they added little NVRAM chips to the boards which hold data about the disk, so you need to make sure they match. I just fried a HDD controller with a bad SATA cable, so I'm having to switch the chip from one board to another to try to recover the data.
Oh dear, here we're talking about age deterioration. Swapping media as a solution only works if the remanence hasn't decayed below the recoverable threshold.
It's certainly inconvenient, but this is my untested understanding of how drive recovery services can work.