So does that level of comfort ever happen with private pilots?
Well, to me it depends on the plane, the flight, the weather and a few other factors. Similar to driving a car.
There are flights (in my case local aerobatics sorties) which I undertake almost as casually as hopping in my car for the drive to work. Prepping for the flight (weather, notams, squawks, W&B, navplan, flightplan) takes less than 15 minutes, preflight another 5, I hop in, strap in, do the checks and go. It's all routine. Just like driving to work - you've done it so often that it doesn't require a lot of conscious thought. Just concentration.
But there are also flights that do take a lot of conscious preparation - in some cases multiple days worth of preparation weighing the different routes against one another, checking and rechecking fuel requirements, fuel stops, PPR, PNR, customs requirements, survival equipment, weather limitations etc. Likewise, there are car journeys that take more forethought. Long journeys in rental cars in foreign countries, at night and in bad weather conditions come to mind.