You can pick up a short but meaningful part of Rt. 66 (called the "National Trails Hwy") via Amboy on the drive between Palm Springs and Las Vegas.
https://goo.gl/maps/ZZHLcBmPZAD2
Including the Grand Canyon and/or Bryce/Zion is probably going to blow your accommodation budget in addition to confronting a lot of people. There's very limited hotel capacity in these areas and it's heavily booked in the summer. Staying in the likes of Flagstaff or St. George isn't likely to help all that much but will be very time consuming in terms of access. Lineups of cars waiting to get into national parks is commonplace these days.
I think the road hours saved by skipping these destinations is more like 20. Remember these are "wheels turning" hours at the speed limit. A couple of trucks or RVs poking along can make a big difference, and stops for photos, food, nature breaks etc. can turn a 5-hour drive into an 8-hour day.