Flatbed versus wheel-lift is decided before the truck leaves, not on scene. All-wheel and four-wheel drive vehicles ride flat, always, so nothing turns unpowered wheels on the road. Lowered cars, a seized or locked-up wheel, and any total loss also go on the deck. A front-wheel or rear-wheel drive car with no drivetrain damage can usually go on wheel-lift, towed by the two wheels that are not driven.
Not everything that strands you is a tow. Roadside covers jumpstarts, lockouts, fuel delivery when a tank runs dry, and a tire change on the shoulder. A winch-out - pulling a vehicle back onto pavement from a ditch, a snowbank, or soft shoulder - is also roadside work, not a tow, unless the vehicle cannot be driven once it is back on the road.
Accident recovery is different work. We stage behind the police unit, wait for the scene to be cleared, and load only once we are told to. A tow you call yourself goes wherever you ask. A tow the police order goes to the impound lot they name, and getting it released from there is between you and that lot, not us.
