Lots to look forward too !
Hello and welcome to RTA !
Your trip looks well planned out and you are in for a real treat ! From Lone Pine I would explore DV and continue to Vegas for the night, giving you that extra night near GC. As you are not heading to SF, perhaps Monterey would be my preferred choice over SC. The Utah parks are great but will make your trip a little rushed as it is. For me the choice would be starting from LA as above, or starting in SF and dropping the coast for Zion and Bryce canyon.
Maybe not southern Utah, but try Monument Valley.
You may not have time for the southern Utah NPs, but if you leave the GC via Desert View Drive, it would allow you to do a detour and see some of the spectacular Monument Valley. No need to do the tour or anything, just driving 160 and 163 to Bluff and return via 191 and 160 back to Flagstaff.
Just driving along those roads, you want to stop every few minutes to take yet another photo. That country is spectacular. (Nothing like driving across the Nullabor.)
Those last three days would look something like...
Leave LV early and drive to GC (a five hour drive, arriving soon after lunch.) Be sure to take in the sunset that night.
Next day up early, take in the sunrise over the Canyon, leave early via Desert View Drive, route 64, it would allow you to do a detour and see some of the spectacular Monument Valley. No need to do the tour or anything, just driving 160 and 163 to Bluff and return via 191 and 160 back to Flagstaff.
Last day, Flagstaff to Phoenix.
You will have to book accommodation in the NP a.s.a.p. Especially Yosemite and Grand Canyon. Accommodation is limited, and much sought after.
Be sure to take your RAC WA membership with you, as it will give you access to free maps and tourist information from the AAA. You could pick up the CA, NV, UT and AZ maps while in LA.
Lifey