I've made three one-way cross country trips in conjunction with work relocations. All were coast to coast and I've taken the southern, middle, and northern routes. I had 2-3 weeks for one way trips. I agree your plan isn't doable but I think you realize that by now. I read the entire thread because I hate when folks post without reading other's suggestions. I was going to offer a similiar plan to busskipper.
Drive hard for two and a half days (including leaving after work) and then see AZ/CO/UT. From the Grand Canyon, you are at least another 14 hours to the Redwoods. Catch Grand Canyon, Four Corners, maybe time for Arches, pick and choose in CO.
Part of my rationale is you likely have seen or can see the Eastern States on another trip. Get to the West and enjoy a new part of the country. In the two cross-countries that I was towing and camping, I found two days hard driving was fine but then I needed a break. We did some half days of driving with tourism and 3-4 days at our National Park destinations. Some will say you can't do something like Yellowstone in 4 days but while working, it beats not going at all.
Have the family list ten destinations in a three state area and then compare lists. Maybe you can visit everyone's top choice and then negotiate from there. I think a day or two at the Grand Canyon is sufficient compared to Yosemite that needs 2-3 minimum (not this trip) and Yellowstone/Grand Teton where 4 is probably a minimum.
If you can get the search feature to work, there is a a lot of info on here from folk's planning posts and trip logs.