It's very possible as I pull up more of the carpet I will find other areas with sections of MDF. It's likely some of the boards were damaged and they pulled em out to level out the surface before installing carpet over it right? Is it reasonable to think I could cut this MDF out and install new hardwood over it? I will then sand and finish it. It's 100% ok if the new hardwood "patches" don't end up looking the same as the old ones when all said and done.
Yes you could repair the hard wood. Check out some more information on line so you understand how hardwood T&G is put down, I helped a friend that messed up quite a few boards to do a repair under a window that had leaked because he did not understand how the toe nailing worked. If the missing area is on the ends of the planks it is much tougher to repair compared to several length wise boards.