We have traveled (from Indiana to Florida) and (from Indiana to South Carolina) over Christmas week for the last several years. We return home on New Years day. We've never had a break down, but we've never had problems find a motel (for the first leg of Florida trips), or a campground (for the first leg of our South Carolina trips). Once you get into Kentucky and Tennessee, campgrounds are open.
Along the way, the roads are jam packed with travelers, truck drivers, and RV transporters driving South. We've traveled on Christmas Day and restaurants, gas stations, grocery stores are open, pretty much every where.
Businesses (like gas stations, service stations, repair shops) make an effort to remain open on the holidays, because they know travelers are out there. They may charge more, but many of them pay their employees more on the holidays to be there for (us) too. They are forfeiting their holiday so we can enjoy ours.
Travel, don't worry about breakdowns. If it happens, just use your head.
See you on the road. FYI, we are leaving for
Santee State Park, South Carolina on Dec 23. We'll stop somewhere near Knoxville, TN Saturday night and then on to Santee Sunday. This will make about 2 - six hour drive days for us. Enough we can enjoy things along the way!