It would take you around 40 hours driving time just to get to Arizona. That does not include rest stops, fuel stops, food stops, etc. So you could plan on 5 days travel each way and still be long travel days.
But with 3 weeks, that would leave you 11 days to visit the sites you want to see once you get to AZ. The 3 you mention are within a day of each other, so you could see them fairly easily once you got there with time to spare.
It would just be the getting there and back home that would be the killer.
Yes, there is a lot to see/do between your home and AZ that you would miss. But if those mentioned are your primary objectives at this time I would say go for it. But plan on the full 3 weeks.