A1314 Magic Keyboard Some Keys Not Working

I bought a used Magic Keyboard A1314 (Bluetooth) and some of the keys are not working.

My definition of "not working" is that when the key is pressed, nothing happens. Here are the list of working/not working keys:

• all lowercase letters work fine
• all numbers work fine
• Using the left shift key, these capital letters do not work: QWERTYUIOZXCVBNM but these do: PASDFGHJKL
• Using the right shift key, these capital letters do not work: ZXCVBNM but these do: QWERTYUIOPASDFGHJKL
• The following symbols do not work with either left or right shift key: <>
• The following symbols do not work: ,
• All remaining keys work

Pairing the keyboard via bluetooth is easy, and I have paired the keyboard with an iMac, a Mac mini, and a MacBook Pro, all with the same results. All the Macs are running macOS High Sierra 10.13.4. Batteries are brand new at 100%.

I have scoured the prior community posts and have tried everything but nothing has worked. I think it is a software issue because of the following logic (just a few examples of many) :

1. Left-Shift-Q does not work, but Left-Shift-P does work, so Left-Shift key hardware is OK
2. Left-Shift-Q does not work, but Right-Shift-Q does work, so Q key hardware is OK

Lastly, I noticed the prior owner's name is still the name of the keyboard, i.e., john's keyboard. I cannot rename or reset this no matter how many different ways I tried (from other community post suggestions) but I think perhaps if there were a way to reset this keyboard to the factory settings, including its name, perhaps I would solve my software issue with the keys.

Any suggestions? I've done the Shift-Option debug and factory reset, as well as holding the power button for 6-seconds. It would be a shame to toss this keyboard in the garbage since it does seem like it's a software issue not a hardware issue. Any suggestions welcome.