Tough to really answer these type questions without knowing more about the family involved, the trip, the temperment, personalities, etc. It is likely different for all concerned.
I did a trip that was basically the reverse of yours last summer. I did not want to have to scramble for spots last minute. We were visiting national parks mostly. I was worried about campgrounds being full. Turns out that not one we stayed at was full. Go figure. I had a 45 day trip with reservations for every night. I did like the security of knowing that.
Forum members offer conflicting advice. The majority will tell you to plan on about 300-350 miles per day (likely 8 hours with stops) before stopping. Then others will tell you not to worry and go to a Walmart in a pinch. So now your at a parking lot at 4:00 PM. What do you do until bedtime. What about the little ones if you have them? What if you have hybrid bunks or slides?
Some people seem very comfortable in a strange to them, remote location, driving around looking for a campsite. Not so much fun if you've been driving all day. Could be dark. Could be raining. Kids could be miserable and acting up. I'll pass on that. Others may not care.
You have to be realistic about how many hours of the day you can spend in a vehicle and how many days of the week you can spend in that vehicle. I never drove more than two days without taking the third day off and just chilling. Trying to cram too much in to your allotted time frame can ruin a trip, IMO. Plus, we didn't want to spend our summer on the interstate. Great for going long distances quickly but not all that scenic or interesting. the goal wasn't to see the country through the windshield. On the other hand you may only have a short window of opportunity that handcuffs you.
What it comes down to is you have to figure most of this out for yourself. Only you know the particulars of your situation. What worked for my family may not work for yours. What works for the retired couple who full time probably won't work for you if you have a young family.
Good Luck!