Private owner aircraft will be either for fun or for going places.
Flying as Onetrack descried will in fact open up light aircraft to people with neither the time or inclination to train for current airways type operation, and quite possibly no interest in flying, other than as a means of getting to their intended destination by the most direct route and when they want to go. Type in or say the destination, confirm you and the aircraft are in agreement, (and here we differ) load cells on either the undercarriage or in the seats and cargo compartment will confirm weight and balance is ok, and you are off. The avionics will confirm a landing slot, so all that those on board need to do at the other end is go to their waiting onward transport.
Flying for fun will be much the same as now, although electric aircraft will be the rule. A fossil fuel powered generator may be carried in a pod if going to a remote destination, although the high efficiency solar cells on the wing, trailer or covers will do some of the recharging, and for weekend fliers, they will ensure the first flight is usually a free one (fuel-wise, anyway).