Well, it is not dissimilar to the drive from Cody to Yellowstone, actually (when last there, there was tremendous road construction and it's not exactly a straight road). If you like driving winding mountain roads for 2 hours or more per day while traveling (and getting back to your lodgings in time for dinner), then kudos on being able to get everyone on board for that. My family would totally mutiny. And we did that drive between Cody and Yellowstone one time, would not want to do it every day for any reason.
That being said, of course, Cody is an actual town, unlike Three Rivers, which is sort of not a town. It is stretched out along about 6 miles of road, with a teensy town center and two gas stations forming the heart of the Three Rivers experience. There is a grocery store. It is about as expensive as the one in the park - but better stocked.
It's too bad you won't get the night time experience of being in the Sierras, though, which is one of the things my kids have enjoyed the most (the bears, the bats, the night birds including owls, the deer). Campfires and ranger programs. Night walks, star gazing (it's amazing up there). Silence a lot of the time.
Three Rivers is hot, dusty and very small. It is not like Cody. Cody has a wonderful museum, many restaurants, a sort of town center, its own interpretive programs and so forth. It is a real city with a Western history and a destination in its own right.
Three Rivers is a place where people stop for gas and a few hardy locals hang on to tiny businesses. Let us know how it goes for you, though. You may well discover something about Three Rivers (we just never stop at the Russian doll store, maybe it's interesting; and the little place with the miniature golf course that my kids wanted to stop at was never, ever open - not once, in 29 years of taking the kids up there). I think it has actually been returned to being a cow pasture, but that the remains of the place still make it appear as if it might be a golf course.
I will say that the We Three Bakery has great stuff and maybe we were starving, but we thought it had great hamburgers. The restaurant right at the bridge is...interesting. It can be very adult-oriented or it can be family-oriented, depending on who else is there...