You would lose a bit of space, but it's perfectly doable with 1 by (3/4) stock.
So with 3 braces along the length, you'll lose 2 1/4 inches.
As I said, I dozen ways to skin a cat. I'd still use a center brace, but you could just attach the ends with cleats. essentially, a small block of wood in the underside corner that is screwed to the outside wall, and then into the shelf.
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Since you know the structure of this recess, it provides even more opportunity.
If you were feeling really ambitious, you could remove the drywall from the inside, and create a full-fledged built-in. Assuming half inch drywall, you'd only lose a 1/4 inch on either side.
I think I see what you mean with the braces/cleats at the side ends. I would only be able to secure this cleat on the very edges perhaps because I don't think there's another stud in between this area of the wall. I might have to drill some pilot holes in different places to know for sure to avoid playing the bigger guessing game. Perhaps I should invest in one of those pocket hole guides as well.
As far as removing the drywall from the inside, I could do that but it would expose all the 2x4s and if I wanted to make it a "usable" surface I'd have to extend the 2"~ border height out probably by 2 times, and I'd lose 2" all around I think.
I'm not sure I understand what you're saying here. In the simplest terms, you'd just be replacing the drywall with wood.
In doing a built in, a shelf unit is built that is pretty much the dimensions of the cavity, usually with a slight allowance that can be shimmed to make sure it will fit. Then the unit is slid into the cavity, and attached (screwed into place). Then trim is usually added to the outside to marry it with the drywall and cover any gaps.
So the built in has only marginally smaller interior dimensions than the cavity with drywall in it. The upside, is that you can build the wood to wood joinery as strong as you wish. You can use housing dadoes, cleats, dowels, dominoes, hell even sliding dovetails if you were so inclined :)
Another aspect to consider, is whether straight wood is the material you want to use. Wood's strength is directional. It is fairly weak at supporting loads when laid flat with the grain. A number of manufactured products seek to remedy this (with their own pros and cons). It would look worse, but a 3/4 piece of decent plywood for instance, would be more rigid in your current setup.