Packs won't show up in game

@PlainSimpleAimie  Please clear the EA App's cache (again), open your game library, click Sims 4, select Manage > Manage Add-ons, and let me know whether your packs are listed as installed or not.  If you see Download buttons, of course click them.

Otherwise, please run the search I described again and post the results.  Even if they're not installed at all, you should get at least one result per pack: the game creates a folder for each one inside ...\The Sims 4\Delta, to hold patch data.  So if you're not seeing those, either you're not running the search correctly (for example not clicking This PC first), or you're not letting it finish.