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. "