360 games on Xbox One X make me enter my password each time

...As it's my kids playing the 360 games, I don't want to give them the password

Before I go to possible troubleshooting steps:

Please do your kids a favor and create separate child accounts for your kids.

DO NOT let your kids play on your account.

Game Progress is tied to the user account, which means, that if for whatever reason you can't allow your kids access to your account anymore, the kids won't have access to their saved games.

For example, some parents have to use their Microsoft Account for "important" things and don't want their kids to access the account anymore, or the kids get older and want their own account. The longer you wait giving them their own accounts, the more game
progress they will lose if they get their own accounts later.

Also, depending on how kids understand their game progress, it might feel to them like doing a crossword puzzle that is already half way done, and every day they play, someone else filled it a bit more.

So in the future, the more they "understand" that someone else is solving the game for them, they might get frustrated and want their own account anyways.

Also, by using your account, your kids are regarded as full grown adults by the Xbox Live system, and you can't restrict what they are allowed to do. In theory they can go and buy stuff with your credit card, play age-inappropriate games, talk to internet
strangers and other stuff that might be dangerous for children.

So please, create separate accounts for each of your children, then go to
https://account.microsoft.com/family and add your children's accounts to your family account.

But now to your Xbox 360 game issue:

It might be, that the profile on the Xbox 360 storage area on your Xbox One became corrupted. Please try to go to the Xbox One settings, and under storage, clear the Xbox 360 storage.

This requires you to download your (and your children's) profiles again (hopefully only once) when you launch a Xbox 360 game after that.

After your and your children's accounts have been properly downloaded, try changing to different Xbox 360 games.

BEWARE! When changing Xbox 360 games, make sure you "properly" quit a running Xbox 360 game.

The Xbox 360 games from "back then" have no idea of the "suspend" feature of Xbox One games. In order for Xbox 360 games to save game progress properly (and prevent them from possibly causing corruption on profiles), make sure you "quit" a Xbox 360 game,
when ending your game session or changing to a different game.

To quit a Xbox 360 game properly, go to the Home Screen with the Xbox button, and on the big game tile with the Xbox 360 game, press the Menu button (the small button with the three lines to the right of the Xbox button). In the Menu choose "Quit", to properly
end the Xbox 360 game, allow the Xbox 360 emulator to save the game to the Cloud and gracefully shut down the Xbox 360 emulator with profile and other stuff to store.

Thanks, do you mean this page? https://account.xbox.com/en-us/Profile/Protection?rtc=1

You need to log in to the Xbox website and go to the privacy settings where you can set it for each console

It might be, that the profile on the Xbox 360 storage area on your Xbox One became corrupted. Please try to go to the Xbox One settings, and under storage, clear the Xbox 360 storage. 

Thank you, that did it! Working perfectly now!

Also thanks for the tips on the family profiles, I'll take a look at those today.

Hi,

You should be able to simply download your profile again and solve this issue.

To download your profile: Push the middle button on your Xbox 360 controller > Push the blue "X" button on your controller to sign out of any sort of account > Download Profile. When you enter in your e-mail and then app password after your profile is re-downloaded
you should be asked to Save Password. Make sure the box is checked and then press A on Sign In.

I also have 2-Step Verification enabled, and I haven't had to enter in my App Password unless I download my profile to another console. Having it so you save the password will make it so the profile automatically logs in each time you turn on your console.
After that's done you shouldn't need to enter in the application password again unless you attempt to re-download your profile or go to another Xbox 360 console.