Can't Play Games on Disc with my Gamertag but can w/ Other Gamertag

I can't play the game on this gamertag using my other gamertag and so close to getting the 100 skill achievement on that and not this lol.

There are no restrictions on who you give your game discs to.

Downloaded games can be played by any gamertag, provided the gamertag that bought it is on the same console.

If by the "Shared with Gold idea" you mean the "Home Console" feature, where an account designates a specific console as their "Home Console", which then shares their Gold membership and purchased games with other accounts on the console, then you already
know how it works:

An account buys games DLC and Gold memberships, and as long as the console is set as this account's "Home Console", all other accounts on that console can play that account's games, DLC and play multiplayer with the shared Gold membership.

If you buy a game on disc, then the disc has to be in the drive to play the game. DLC that you get via a code, is bound to your Gamertag, and through the "Home Console" feature, others on that home console can access the DLC content as well (usually. I have
heard of CoD Season Passes being ONLY for the individual that bought them, and aren't shared, but I don't play CoD, so I can't verify that).

You can switch the underlying Microsoft Account email of your Gamertag, and since the purchases and code redemptions are tied to your Gamertag, you will keep your purchases.

But you shouldn't create a new Gamertag. Use the Gamertag you used on the Xbox 360, since your past purchases on the Xbox 360 can then be accessed through the Xbox One Backwards Compatibility. Only change the Microsoft Account email that you use to sign
into your Gamertag by creating an alias.

My game will boot w/o the HDD. I don't have a gamertag or anything, and can't play LIVE, but campaign works. The only thing left on the HDD is original BO. I moved my tag to a USB thumb drive. I'm tempted to format the HDD to see if it fixes it. I don't
understand why all the other games work fine, except this one!

You cannot have two Home consoles under one gamertag. However, what you can do is download the games that you've purchased digitally to the second console. You will be able to play those games on the second console with no issues. Others can play games
on your current console (Home console) without needing your gamertag signed in.

This does not work with disc based games. If you have disc based games then you would need to purchase a second copy.