Xbox one will not start any digitally downloaded games.

Any digital games or DLC you paid for are tied to the account that paid for it and the gamesaves are auto synced to the cloud.

I've got an Xbox One X and Series X and I have the digital games on both of them.

As for the gamesaves, whenever you start a game on your new console the game will sync with the cloud to download the gamesave.

with the xbox one all game saves are loaded to the cloud through your account even if you don't have xbox live gold so when you download your profile to any xbox including if you get the new xbox one x then any digital games you own will show up and game
saves will be synched with the cloud game saves when you start up the game.

Every game or app you digitally own will show in your ready to install section of my games and apps. Just find the game you want to download and start it. If you have a disc though, then you will need to insert the disc in the consoles drive and start
the install. If you have the digital only Xbox One S, then your only option is to purchase the digital version again to play it on there.

I bought a disc copy of Batman: Arkham Origins a backwards compatible Xbox 360 game and I used to be able to see it for sale digitally on the Microsoft store, but ever since I installed onto my xbox via disc i can’t see the game itself in the store,
I can only see the add ons. Any reason as to why this is happening? I’d like to own it digitally instead but the digital game page is nowhere to be found.

What you mean with "for sale digitally on the microsoft store" ?

If you have the original disc in your tray, the console start downloading the Xbox One version.

any day one release games should be available for digital download on day one.