The trays that are only for sorting look like this

https://www.amazon.com/Becko-Stackable-Sorting-Accessory-Hexagonal/dp/B07Y21LKWQ/ or this https://www.amazon.co.uk/Clementoni-37040-37040-Puzzle-Accesories-Multi-Coloured/dp/B07MXXLQSM/

There isn't enough room inside those trays to actually work the puzzle so they're strictly for sorting purposes.

The puzzle boards/trays for doing puzzles would be the long flat ones (or those ones with the lip that I mentioned), and that's where you do the puzzles.

If you're talking about the Dowdle page of puzzle trays, they are great for doing puzzles on I think -- they apparently do make mini puzzles but they also have larger trays like this where you could do bigger ones

https://dowdlefolkart.com/collections/puzzle-trays/products/large-puzzle-tray

or this (the XL one, which would hold 500 piece puzzles but probably not 1000s I think) https://dowdlefolkart.com/collections/puzzle-trays/products/extra-large-puzzle-tray

So if you were going that route, I'd consider buying a set of trays like these https://www.amazon.com/Becko-Stackable-Sorting-Accessory-Hexagonal/dp/B07Y21LKWQ/ (you may be able to find 'em cheaper, just an example) along with a puzzle board/tray like this https://dowdlefolkart.com/collections/puzzle-trays/products/large-puzzle-tray

Then, you could pop a pillow underneath, and I think that setup could work well.