@Foxley828 For these households that showed up, let me make sure: were they loading before or not? I thought not based on the timeline you posted, but if they were working, please try a set of files, all with the same timestamp, from before you transitioned to the App and from the time period where you can no longer see your files.
There are two distinct possibilities here. One is that you simply have too many files in Tray, and the fix there is to be selective in what you load. As you've seen, you can swap these files back and forth as you wish, so you could set up different Tray folders with different sets of content. It might take some time to organize everything, but once it's done, moving the folders around is simple.
The other possibility is that some of the Tray items are corrupt. With the second set of files you tested, the list of files suggests that there were several different saved items, but all at the same time, so I'm wondering whether you pressed the Save button over and over, or the game reacted as if you had. If it's really just the one household, I can help you get rid of the duplicate files. But it's also possible that some of the files don't work at all, like there should be several copies of this household showing, but most do not because the files can't be read. And if that's the case for other Tray items, ones where intact duplicates don't exist, there's no way to save them.
Point is, please test some older files that you know weren't showing up when you reported this, and let me know how it goes.