@RunaAster The file should be in Bin as well as Bin_LE. Please check there too and make sure there's a copy there.
Now if you want to fix the unrecognized card issue, open the copy of graphicscards.sgr that's now inside the Bin folder. Use crtl-F to find 5916, change it to 5917, save, and quit. Then check the config.log file that's in Documents\Electronic Arts\The Sims 4 to see if your card is now listed as found and supported. This is the same approach I mentioned here:
https://answers.ea.com/t5/Technical-Issues-PC/PLEASE-READ-Unrecognized-Video-Card/m-p/8882426#M169532
But don't move any files around while you're doing this, just open the ones mentioned.
You don't strictly need to complete this step if you don't want to. When you're done with it, or have decided to skip it, then yes, you can move the contents of Saves and Tray from the old Sims 4 folder to the one that's now in Documents\Electronic Arts. Your saves should be back, as should any builds or households you've saved to your in-game library.
It's a good idea to copy the files rather than dragging and dropping them: this way, you'll have a backup copy of everything if you need it. You can highlight the whole set of files, right-click on them and select Copy, then right-click in the empty space where you want them and select Paste.