Amtrak isn't awful in business class. Problem is that there isn't much good schedule coverage for trains, except in the DC-Boston corridor. Only in these areas will there be any kind of pedestrian-friendly sight seeing.
I think a private car is the way to go. Maybe even a smallish RV or pull-behind would be good if you don't mind staying at campgrounds. You'll save a lot of money by using an ice chest and little grill for food preps. These can be found for a song at just about any thrift shop.
There are a lot of nice stops along 95, with decent nearby campgrounds. Savannah, Charleston, Saint Augustine areas are all worth a stop and campground stay. Richmond/Petersburg area is fantastic for visiting Civil War sites, if you're into that. DC is great for museums. Lancaster County, PA is worth a drive by.
Anastasia State Park in Florida (near St Augustine) is fantastic for self guided hikes, bird watching, etc. https://www.floridastateparks.org/park/anastasia
I have done a lot of urban camping (in my car) by staying in parks areas. There are a lot of state and national parks around, but some require camping reservations. Anastasia was (I think) $12/night per vehicle when I last stayed there.
I would advise a family traveling this way to arrive to US with basic comfort clothing and cash. Stop at thrifts along the way (salvation army, goodwill, etc) to pick up any camping gear or beach toys as might be needed. You can donate it back to a thrift or to a needy family on the way out of the country.