Summer is a vague term, are we talking June when things are a bit slower or July/August during peak season when just about anything worth stopping at are crowded?
IMHO, 6 weeks is doable, but really not sufficient time to see enough to make it worth while.
Drop New Orleans, Smoky Mountains, and St Augustine. Those can be done on long weekends or single week trips.
Unless you have some special reason, I would also skip anything west of Grand Canyon due to the time. Instead, hit Zion, Bryce, Canyonlands, Tetons. Less driving and more sight seeing.
Maximize travel the first three days out and last three days back home. This should put you 1,500 miles out figuring 10 hours / day averaging 50mph.
In 2005, we made a trip similar to yours hitting Grand Canyon (3 nights), Bryce (2 nights), Zion (2 nights), Kings Canyon (4 nights), Yosemite (2 nights), Oregon coast (4 stops 2 nights each), Glacier (3 nights), Yellowstone (5 nights), Pipestem(1 nights), home. A few other small places along the way once we made NM, but otherwise was moving about 200-400 miles/day which left a little time to sight see. Luckily, we had 8 weeks to do the trip, fun but very tired when we got home.
Plan on arriving back home with at least one day before going to work. Buffer in case you have problems plus time to put the house back in order.