Need advice: should I open the back of a 2013 Retina MacBook Pro as a means to recover data?
First, some (semi) good news. Your data is most likely fine. The $1000 option is for severely damaged and / or corrupted drives.

The SSD adapter is the best choice - it will turn the SSD into an external drive and you can copy stuff off in the usual manner. 2 weeks is unreasonable - do you live somewhere not serviced by FedEx overnight?

Putting the old drive in a new MBP is possible - it will either fit or not fit, at which point you wait for FedEx to deliver the SSD adapter. However, the new MBP will require the most recent OS. If you kept the old one up to date it should run just like the old one. If you were still running Yosemite or Mavericks it will probably not boot - the old software doesn't know about the newer hardware.

A quick look at teardowns (you didn't specify 13 inch or 15 inch) show the drives are somewhat different - when I say "fit" above I mean the connector goes in the slot far enough to work. Not that it sits in it's space and you can put the backplate on again.

If you can find a 2013 MBP from any of the used computer dealers you definitely will be able to drop the old drive in and carry on as before. Updating your backups (and backup process) first of course.