You should always book assistance via your airline,

Normally they will provide the golf carts since they can move several people at once but it is possible they may provide wheelchairs
Even if you have your own mobility scooter it may be checked through to your final destination and may not be brought to the gate.

While your father may wish to maintain as much independance as possible it is better for everyone to take full advantage of the airport assistance services.

Airports are full of people in too much of a hurry and they may not always give as much time and space needed for someone who cant move as fast