Fewer places you can actually see.
As mentioned, if you were going to devote one of your days in LV to the GC, then plan to spend it in the Grand Canyon on the way. Staying within the NP is highly recommended, but maybe too late to get reservations. You would need to reserve accommodation near the GC, Tusayan being the closest. (There is also a great IMAX movie of the GC in Tusayan.) If you want to do a flight from LV, by all means. I did, but do it in addition, and not instead of visiting the NP.
If you were to drive from NOLA direct to Dallas, via Houston, on to Amarillo and west to Albuquerque, what in particular would you miss that you (and others) have your heart set on? From Amarillo west I-40 is incredibly scenic. Take whatever opportunity you can to take in the vista and capture it on film. There are other attractions along the way, much closer than the cities you have mentioned. The Santa Fe area would be one. If time permits, you could experience some of the minor roads by taking 104 from Tucumcari to the New Mexico Las Vegas and I-25 into Santa Fe. In Albuquerque there is the Sania Cable Car trip, the oldest church in the old town and lots of history. And of course you could stop in Winslow AZ, and 'stand on the corner'. When you get your maps, you can see the rest for yourself.
As Buck mentioned, if you were to visit all the places mentioned, you would not have the time to spend any time at any of them to sight see. Your time would be spent in the car, virtually the whole way.
Remember you will be driving through country like you have never seen, and will never see at home. Make the most of it.
Lifey