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I think the easiest way to collect the Lego pieces from the floor is by using a vacuum cleaner.

Then look at the sorting as a separate problem :)

We have a “Lego sheet” - just a regular bed sheet that goes on the floor before playing with Legos. When it’s time to collect, you make a “bag” with the sheet, grabbing (most of) the pieces, and then we “pour” them into the final container. The sheet goes in top of the pieces on the same container so it’s also the first thing that comes out the next time.

I guess this was a common enough solution that now you can find products that are essentially what you're describing, e.g.: https://www.amazon.com/SAM-MABEL-Storage-Basket-Play/dp/B0BV...

From the item description: "TIDY UP IN SECONDS: Say goodbye to messy playrooms with our storage organizer! The play mat provides a dedicated area for creative play, and when it's time to pack up, simply gather the handles and tip everything back into the compact storage cube. "

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