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Some data points...
Everybody has different wants and needs. What works for me might not work for you, my limitations may not be yours.
These days my wife and I like to spend 3 nights at most of our stops, with occasional one-nighters if the distance is too long between places we want to see.
In 2013 we did a "Northern Parks" loop; starting at Grand Teton and heading north as far as Jasper, then back home (SF Bay Area).
Here's the stats: 26 days, 10 stops, 11 'transit' days, 3,500 miles (average 320 per transit day). Did not feel rushed at all. A couple of the days were pretty long, up to 500 miles or so, but we try not to put days like that back to back.
In April we're planning on a "Deep South" road trip: start in New Orleans, end in Atlanta. That is looking like: 25 days, 9 stops, 10 transit days, 1850 miles (200 per transit day).
With three weeks and some planning 3,000 miles should be easily done, without needing to spend 10-12 hours behind a wheel.