I've done two cross-country road trips.
First was three weeks: NorCal down southern US and over to the East Coast, up the coast to NYC, Niagara, then Ohio, over to Chicago and then the Dakotas, Yellowstone, then home. We hit up a few theme parks on that trip and landmarks/places of interest.
Second was 5 days: NorCal to Ohio for me to move out for grad school. Purpose was not for theme parks, so we didn't hit any, but did do STL City Museum.
One thing I will say about these types of trips is not to underestimate the amount of time you will spend driving. Three weeks will go fast.
Picture is somewhere in Nevada, I think. It's not impossible to hit a park every other day but it is improbable as there are a lot of states you'd have to go through that have a whole lot of nothing.
Looks like if you chose the northernmost route, you could hit up maybe Lagoon in Utah (EDIT: Lagoon is also out of your way for the direct route), Six Flags Great America in the Chicago area, then do Cedar Point and Kings Island in Ohio (KI is out of the way but worth it). Of course hit Kennywood when you're in the Pittsburgh area. I'm not sure how far east you want to go after Pittsburgh, but Knoebels in Elysburg, PA is a great park if you can swing it, and Hersheypark in Hershey, PA. There are others further east and south that are great, but then you're going even further out of your way and adding more drive time.
For the trip back, if you take the southernmost route, you could easily hit up Worlds of Fun in Kansas City (but if it's a "good park" is TBD). If you want to detour, Silver Dollar City in Branson is great and you could maybe swing Holiday World, but it looks like both of those would be away from the direct route.
Really, you need to do a day-by-day plan and figure out how much driving you want to do every day, and factor that in with the length of time you'd have available to take detours to theme parks. You might be better off just flying to PA and renting a car to hit the parks in the NE.
Also, RCDB is a great resource to figure out which parks might be near locations you will stop at.
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