How to transfer games from hard drive to series X

Not really...

Xbox Series X games / upgrades / updates has to be installed onto the Series X internal or expansion card SSD,

You can have Xbox One games on your external hard drive that will get an Series X upgrade but the games will need to be transferred to the Series X and then the Series X upgrade will need to be downloaded onto the Series X SSD,

As an example...

I have Marvel Avengers on my external hard drive that will get the Series X upgrade and I will transfer the game over to the Series X SSD ready to download and update the game to the Series X version when available.

If the games are Series X optimized, you will need to transfer them to the internal hard drive on the Series X in order to play them, but as Obsessive Power stated, you can just plug the hard drive in and not have to redownload anything.

Then yeah, you can plug it into the new console and use it like that or transfer from the hard drive. Any games that are optimized for the Series S/X will not run from that drive, but you can store them on it. So you may have some games that will have to be moved to the internal drive on the Series S.

Yes they are all digital games. Maybe the problem is with the profiles. My husband left his profile on the old Xbox and let my son have it. He created a new profile for himself and I kept my old profile. The only thing that doesn’t make sense is both my husband and i could play the games on the old Xbox, even if his profile was logged out completely. I feel like the profile or console shouldn’t matter if the games on owned and on the hard drive.

The game purchases are tied to the account that made them. Using the old account that was given to your son, sign in on the new console and have that old account set the new console as "Home". This will allow that accounts content to be shared to the new console. Only set the console to Home with that account and not others. Once the account has set that console to Home, the account can be either removed from that console or you can turn off auto login for that account so that the two consoles aren't trying to sign in to the same account.

Do the games not show up in the games library of the new console when the account that purchased them is signed in? These are all digitally purchased games and not disc based installs, correct?