In my opinion, It's not so much a driving issue, as it is a parking issue. The bigger you are the harder it is to park, especially at the smaller sights.
One way to beat that, is get to the sights, as early as possible. Things really heat up by noon, one o'clock ish. In terms of crowds
Just my opinion but the park campsites, are not really all that great, the tend to be small sites, and the CG's are crowded. On the flip side you do have the occasional elk herd wander through.
You say you are a fancily of 5, I am guessing three kiddos, you might consider staying outside the parks, at commercial campgrounds. As they tend to have things kids like, as in pools, wifi etc.
Sounds like a wonderful plan and trip though, and a great family adventure
