The idea to do it in an airplane is a combination of:
1. Save travel time.
2. We all love to fly.
However, the emphasis should not be on flying but to see USA. I would say, 30% to focus on aviation, 70% to focus on the monuments.
The hard part will be to decide what to include and to what level or depth. Three weeks is not nearly enough time.
Figure two days for seaplane rating - you can do it in one if everything is just right. Maybe just do a few intro flights instead. Seaplane ratings in upper Midwest (Minnesota, Wisconsin, Michigan) or Florida. Where will you put mother and son while you get rating?
Glider transition not nearly enough time - do a couple of mountain or ridge soaring flights dual. Again, where to entertain others while you're soaring? Great soaring around mountains, so Reno, Marfa, Pennsylvania and many others. Yes, you can thermal in Iowa but.....
All can go up in a balloon somewhere or ride in the Ford Trimotor or B-17 at OSH.
A 200 knot airplane is going to be a challenge. It will be expensive and rare. It will be fairly hard to even find a C182 to rent. You'll have a lot of baggage to haul around.
Fly into Seattle and spend a day. Take the Amtrak sleeper train through the Rocky Mountains to Minneapolis and spend a day or two there. Do some float plane flying (could fly floats in Seattle, too). Rent a car and drive to Oshkosh for a couple of days. Drive on to Chicago for 2-3 days. Get someone to fly you around the area. Drive to Dayton OH to aviation museum. Finish trip in NYC.
Alternative is to fly into San Francisco and come east on I-80 as Ron Levy suggested.
What you miss on the above two routes is the Grand Canyon and the American Southwest desert, which are pretty impressive.
Another option is to fly into Miami, come north west through Atlanta, St. Louis to Oshkosh and back east through Chicago and NYC. Great culture and nice scenery. Plenty of opportunities for soaring and seaplane flights.
American trains are nothing like Europe, but they are a consideration. You'll probably find some legs best done by renting a car.
Finding a place to stay for your mother, your son and you at OSH is not trivial. Plan to stay 40-60 miles out and commute in.
There will be other, probably better suggestions. All will have their merits.
Again, let us know where you decide to go and I bet many of us will be happy to offer you a local ride in our plane.