I meant to say risers are going to be birch plywood not oak

Those treads are solid wood, they are just pieces glued together with a top and bottom laminate layer.Seems to me they are using lesser quality(appearance) wood. Probably does not impact strength, but every joint/laminate layer provides more opportunity for later problems, especially with stairs and the stress they take.

I always used treads from a real lumberyard, those are generally the same width/length(with two different lengths available to minimize waste on more narrow stairs), but are 1&1/8" thick and made by gluing pieces together.

That way, dimensional lumber can be bought and machined(jointed/planed/bullnosed) to make a matching landing. For your landing, I would get enough dimensional(boards) oak to glue together to fit the space, orient the pieces for the grain to run inline with the stairs(leaving the edges the long sides). Then bullnose the two edges.

To do all that requires some machinery. Either find a woodworker/cabinet company/etc. that has the necessary machines or have the pieces machined by the lumberyard(home improvement stores do not do this type work).

The machines are: jointer, table saw, router and if the boards are too thick, a planer. Basically, the table saw cuts to length and width(if necessary), the jointer makes a 90° edge for precise gluing, the planer shaves off wood to make it thinner, and the router(with a round over or bullnose bit) creates the edge profile.