OP, you're in a difficult situation right now, but I think you're getting a lot of good advice here. Since you live in your house and see how the things on the floor came to be there, you're more used to them and are able to distinguish good things that haven't been put away, from trash that should be thrown out. To a more or less complete stranger just walking in, who isn't familiar with your possessions, I'm sure it was quite difficult to see that there were good things mixed in with bad. They could
have gone the route of picking up every single thing from the floor and presenting it to you, so you could tell them what to do with it (if you were available); but that would have taken a really long time, and not been an efficient way to get the whole house clean. I'm sure they were thinking that if they indeed literally shoveled all the "trash" on the floor into bags and got rid of it, it would be a huge help to you--a "clean sweep" really.
Also, you said you couldn't figure out how to tell them not to do this once you realized what was going on. I do understand that you haven't been well, and you weren't expecting them to act this way. But if you had said, "Wait a minute. Not everything on the floor is to be thrown away. Could one of you bring that bag over to me on the couch and help me go through it?" I hope they would have agreed. I'm not sure how they would have known this is what you wanted otherwise. Something like the toy sorting piles should have been easy to explain--"My kids have been making piles of toys they want to keep and toys to get rid of. Please don't throw away anything from this area here," or "Please let me know when you get to this area, so I can guide you about what's to be thrown away."
I know this has been very upsetting for you and I hope I'm not just dogpiling. I feel like you've gotten used to your situation in some ways, but new people just coming into it, with the stated intention of cleaning, instead saw a chaotic mess and they reacted accordingly; and you didn't seem to be effective at guiding them, so I'm not sure how else this could have gone.
I POD the suggestions about getting DH and the kids to help. I know it's much easier to say that than it is to do it. But maybe this situation is exactly the catalyst you need to explain it to them. "The kind ladies who offered to clean the house threw out a lot of your toys/possessions because they couldn't tell them from the trash that was also on the floor. From now on, things must be put away at the end of the day, or they might be thrown out." Perhaps you could at least find some cardboard boxes to put on the floor, so toys and game parts can be corralled in there.