Sims 4 suddenly needs to be downloaded again - but won't

@Fesja  When Sims 4 is still installed on a computer but Origin thinks it's not, the most common reason is that an antivirus quarantined TS4_x64.exe, which runs the game.  So please check your antivirus settings and add any exceptions necessary.  If this doesn't help, please let me know what antivirus you use.

For the last played date changing, it sounds like you may not be signed into a different Origin account than before.  Please double-check your purchase history on ea.com to ensure you're loading the correct account.