Not that much difference in miles/locations
I don't see that much difference between your two trip ideas. Except for dropping off Boston and Chicago, they're about the same.
You can figure that it will take 5 days to drive from one coast to the other. So, with 3 weeks, you have plenty of time to explore along the way.
That said...it sounds to me like you need to work through the quiz that Michael linked you to with your friends and determine what type of trip you want and what you want to see the most. Of course, not everybody is going to get to see everything they want and some compromises will probably have to occur. But if you remember that there is no one right trip or best trip and that whatever you choose will be new and fun and exciting, it makes the choices easier to make.
It seems like most of the places on your list are in the western part of the US. Have you considered just doing a loop of one part of the country instead of a one-way, cross-country trip? This would give you more time to explore and would keep costs down. Roundtrip flights and car rentals are usually cheaper. However, if budget isn't an issue, this may not matter to you.
I think you need to get a good US map and look it over with your friends, marking the places you all agree that you want to see, and then seeing if the driving portions are feasible in the time you have. I think alternating cities with natural wonders is a nice mix to shoot for. Sounds like you have friends that like both so that should work.
Anyway, since there's no one best trip, continue to discuss this with your friends and come on back here once you've come up with a bit more ideas to get some feedback from us. We're here to help but you need to do your homework, too.
Also remember that either route will take you through small towns and "real" America.
Interstates vs US highways
Taking US-50 would definitely put you in the mountains, but as you stated, it is going to take you longer to take the trip. I-70 also gives you some spectacular views of the mountains.
But if your primary concern is time, definitely take I-70.