@eliii_b @Freydaleah @Tremayne4260 I managed to install the game using a backup installer from my Windows 10 laptop. The installer was downloaded directly from Origin at the time and now I'm glad that I saved it. Once installed, the game ran normally even with Origin closed, and I got the Update button when I logged into Origin. As expected, the Play button is greyed out and the update fails to download.
The installer (if Origin didn't delete it) should be located in C:\ProgramData\Origin\DownloadCache\, so look there if you're trying to reinstall it.
However, I am still able to run the game by launching it directly from the executable. Look for MySims.exe in C:\Program Files (x86)\Electronic Arts\MySims\bin\ and see if you can launch the game by double-clicking on it.
My guess is that EA is preparing MySims for the new EA App, which could explain the sudden Update and Locate Game functionality. As far as I am aware, only games that are completely managed by Origin have that, and MySims would need to ditch the legacy installer to be compatible with EA App. This seems to be an issue on EA's end, so I'm afraid we will have to wait for them to fix it.
EDIT: Anyone else having the same issue, please click the Me Too button on the original post so it has a chance to get more attention.