Plywood seems like a good solution here until you look at the material in detail. With modern plywood in particular, in addition to generally not being as well-made as plywood used to be (more voids, junkier wood in the core layers) over the years the surface ply has gotten incrementally thinner, to the point where on some plywood it's now like a sheet of paper.... and about as durable!
Even very good quality cabinet plywood (note: not all "cabinet grade" plywood is good) has a fairly thin surface ply of the 'show wood' and wouldn't be made to have the necessary durability. What is made to be durable enough is flooring, where regardless of what's used underneath the wood or bamboo laminate surface is significantly thicker than that on plywood.
So for a high-wear surface if the choice were between plywood and flooring I know which one I'd go with. Since you can't just use the flooring by itself it would need to be mounted on a board material (plywood or OSB) which will make for a very thick 'mat'! I don't know how high the ply is on your carpet but it would need to be in the range of 70s shag carpet for the resulting thickness not to be some issue :-)
Whatever wood/bamboo solution you go with the ideal finish is the same thing you'd use on a floor. A quality floor-grade varnish should give you the same durability you'd have if the carpet weren't there and you were rolling the chair directly on a wood floor.
What about the chair?
There are two things interacting here and you shouldn't look at just one of them. I think you should also examine the castors/casters on your chair.
Castor wheels are made in various materials and harder ones are very wearing of floor surfaces. Modern laminate flooring (which has a very thin resin-reinforced paper surface) wears quickly when hard castors are used, but much better if softer ones are fitted. Regardless of what mat you come up with if the chair's castors are hard it would be worth changing them to a softer type.