I bought the game, uninstalled for disc space, and now I can't reinstall, says "buy game"

As long as you are logging into this battle.net account in the desktop app you should have access to install and play Diablo II: Resurrected.

Double check the email you are using to log into the desktop application and try changing your password. That should force your account data to update and show that you should have full access to Diablo II: Resurrected. If that does not work then the following should clean up any bad app data:

Let’s manually clear the Battlenet Desktop Application and all supporting data from the system. This will not remove existing game clients.

1. Open file explorer/file manager (Right Click Windows Start)
2. Navigate to the folder where the desktop application would normally be installed. Delete that folder if it exists (usually program files x86)
3. Type the following into the address bar (not the search bar): %APPDATA% and press enter.
4. Delete the Bnet and blizzard folders inside the appdata folder if they exist.
5. Repeat steps 3 and 4 with %PROGRAMDATA% %LOCALAPPDATA% and %TEMP%
6. Empty the recycle bin.
7. Now reinstall the app to the C: Drive.

Create a new administrator account: Creating a New Administrator Account - Blizzard Support

Once the account is created, shut down the computer, wait 60 seconds, restart, log into the new profile and try the app/client.