This should be a piece of 3/4" hardwood in my opinion, not plywood.
I got into the habit of using plywood in place of hardwood in certain situations, and I'll tell ya why. In this situation, the add on transverses four (4) intersections of wood perpendicular to each other. Hardwood may develop a twist or bow that could distort the joints its there to solidify. In actuality, the add on piece just creates more rigidity to those intersections which are supports for the drawer box openings.
Any weight placed on the top of the counter will distribute across OK, but I wouldn't stand on it.
I would stand on mine. Even with frameless cabinets, the top rail under the countertop is stout enough to support the top. When I deliver those base cabinets without the top installed, they get picked up by that rail, and/or other cabinets are packed on top of the rail. When installing, I install the base first, and without a top, I lay on a loose piece of ĺ" ply and stand on it, holding the upper cabinet to install.
Now as to the gap that will be left by removing the center post, you can either reattach it to one side, rip in in half and make it symetrical, and attach the 2 pieces on both sides, or just attach it to the back side of one of the doors. Yes, you'll have to open the other door first, but small price to pay and much less work. Maybe she will throw a fit tho, I donno?
That would be an acceptable fix to take care of the gap due to the missing center stile. Done that way it becomes what is referred to as an "astragal". In looking at the photo, it appears there may not be enough width of the stile to attach one piece to one door, or split it and attach a half to each door. If there is enough, you could have a minimal gap in the center of 1/16"-3/32", and then each door would open independently. Worse case scenario, you may have to make a wider piece, or pieces.