Diagnosing possible failing drive to see if it's worth reinstalling macOS
I believe you're overlooking the obvious:


  Removed drive, attached to another Mac via USB/external enclosure
  
  
  Disk Utility sees the drive but no volumes under it
  Mount button is disabled
  First Aid says "Problems were found with the partition map that might prevent booting"
  


Remember, First Aid and fsck can only fix logical problems (filesystem level), they cannot fix physical problems with the drive itself.

Your drive is failing.  

One of the tests that should have been run was to get the SMART status of the drive in question (you cannot do this via USB)

$ diskutil info disk0 | grep -i smart


If it comes back as anything but "Verified," you've got a failing disk.  This test however, isn't the end-all.  It's good for a quick diagnostic, but may give you a false positive.  Even if you get "Verified" you'll still want to run a disk utility like Disk Drill or Disk Warrior to do a deeper scan. 

Now, given that this is a 2011 MacBook Pro, it's a coin-toss as to whether you should spend money fixing it.  A new Samsung EVO SSD can be had for about $60 USD, but no matter how inexpensive the drive, you're still left with a 9+ year old Mac.  

Personally, I'd look at either finding a super cheap SSD and resurrect it for knocking about, or I would part it out (worth more on eBay as parts than as a whole) and put the funds toward a new machine.