I'm a btrfs noob, so I'm skipping any tools that may fix the disk, try those first.

If the media isn't dying, try a file carving tool like photorec. Idk if testdisk supports btrfs but it's worth a try, it's my go-to for undelete and finding lost partitions.

Edit: missed most of your post. First you need to check the kernel logs using the dmesg command. Look for errors that may explain why the disk doesn't show up, especially if it lists scsi or sata in the message.

Edit again: you may want to check the disk's self reported heath using "smart" data. Many bios menus show this info, and there are programs to get it on Linux. If there are too many read or write errors, you need to decide how important the data is. Professional recovery can probably get all of your data if you stop using the drive now and send it in. DIY recovery using a file carving tool would work best if you have another disk to make an image of the failing one with. ddrescue would be the tool for the job to create the image. If you don't have another disk large enough, and the files aren't super important, you can run file carving on the failing disk directly, but the more you use it the greater chance the disk will corrupt more data.