Interesting idea.
This could be quite cool if you laid the 'sheets' down in a creative pattern but then you would need to figure out a way to connect the pattern so it interlocks.

It may be more aesthetically pleasing, less expensive and more of a DIY'er project to put down a dry laid flat stone or concrete paver patio ontop of a sand base.

If you are going for a more modern approach large square and rectangle pavers will work. Or if going for a more rustic or traditional look large thick pieces of bluestone could work.

You have a variety of options in how to finish the edging depending on on how you space and back fill the joints and seam the edges.

Try looking at the website Houzz. type in the search bar : dry lay stone patio and see what comes up to inspire you.

It sounds like a permeable surface is going to fit the bill due to the tree roots.

photo below shows a modern dry lay 3x3 concrete paver path surrounded by black river pebbles and seamed at the edges with metal edging.

Agreed, the boardwalk is an interesting idea, but I think it will be expensive and probably not practical for an area application.

It would be helpful to see a picture of the subject area and some space around it to provide context. I have an idea that might work, be inexpensive and relatively easy to do ... if you're willing to try something experimental. It would only work for lightweight type furniture.

The boardwalk looks great as an access path over the top of an existing landscape. You could lay large pieces of flagstone or slate for the seating area to have a stable place for chairs that the dogs or guests will not fall off of the edge of or get stuck in cracks. There is also a new pressed rubber interlocking square that can be cut to shape to create a malleable surface for seating that is rather impervious to weather. Big box home improvement stores have them in black or rust in a couple different designs. There is a garden in NorCal near the beach here where I hike that has had boardwalks through their garden with potagers planted in the open areas. The decked boardwalk has held up really well and weathered nicely over the years. They are common in the east to cross dunes to the beach and here in the west as woodland paths and for crossing marshy areas. Use treated wood or composite if that's the direction you want to go. Best bang for your buck.

You're essentially talking about building a ground level deck. Check with your local permit authority but around here, that doesn't need a permit. Rather than shoehorn in a product that wasn't quite designed for the application, or put in a patio that's going to heave because you're in Mass, I'd just do a floating deck. Either set the joists on those floating deck blocks sold at the box stores, or lay down 6x6s for the joists to bear on.

deck tiles?