Have you tried having a family meeting where each person presents his/her concerns and then you brainstorm solutions? Here's how my mom got me to put away my shoes when I was 6
I relate very much to your problem. Starting when my son was 2, he and I were out of the house for many hours a week, as I increased my working hours and his attending preschool increased our commute time, and meanwhile he was developing tendencies to adopt various ordinary objects (often things other people were discarding) as toys and to play in ways that involved arranging objects over a large area of floor. He still has these tendencies, and the clutter was a nightmare until we finally found several at-least-somewhat effective approaches:
1. Provide kid-friendly storage, using containers that are easy to tell apart. We got one of those grids of shelves with the fabric cube drawers, each cube a different color, and had him decide what kind of toys go in each cube. Also, for some of his toys that have a lot of pieces (like the wooden train and its tracks) we use fabric drawstring bags that sheet sets came in; we can hand him the bag and say, "Pick up all the train parts."
2. Take 5 minutes after he's in bed to confiscate the objects that are bothering you most. I'm often surprised at how much better I feel after getting to choose what will make the biggest difference to me and converting that mess into a clean space! For a while we used a Treasure Chest system in which we rewarded a day of good behavior by letting him choose what to reclaim from the box of confiscated toys. Mostly we've just waited for him to notice it's missing and ask for it, at which point we talk about the need to pick it up when you're done before we give it back...and if it's persistently left out, or he never notices it's gone, it gets given away.
3. Make specific requests instead of just telling him to "clean up". Again, getting to choose what area will make the most difference to my sense of peace helps me feel better. For me and my kid, "Put away everything on that table" works much much better than "Put away stuff for 15 minutes" or "Put away 15 things"--he will work slowly or find the tiniest possible objects to put away, and that's really irritating!
4. Set an example: When I am picking up something of mine, I kind of think out loud about how I approach the pile, decide where to start, sort into categories, find appropriate containers....
5.Give him responsibility for cleaning a surface and explain that the first step is putting away everything on the surface. My son likes to clean the tabletop. In order to do it, he needs to put away the clutter he's left on the table. We've had to insist repeatedly that it's not okay to move it to his chair and then put it back on the clean table--he has to actually put it away.
6. Use a token or sticker-chart system to reward him for the times he cleans up the first time we ask. Not only does this reinforce positive behavior for him, but it shows us that he's not as bad as we thought--the times he did what we asked become more memorable.
For us, the "help him pick up" approach has gotten less and less effective because we have the kind of kid who loves to debate and argue and order us around. When we are already stressed by the mess, pressed for time, and annoyed with him for making such a mess in the first place, "NO Mama, put the square blocks at that end of the box and only rectangles over here!" is just infuriating. I guess fundamentally, we don't want to help him pick up; we want him to help us by picking up, and we resent feeling like we're helping him and thus at the mercy of his ideas about how to do it (which are almost always inefficient, taking longer to get things less clean). So I totally understand why you're feeling you don't have time for that!
I hope some of these ideas help. Good luck!