Technically correct, for a room fan, though the central A/C fan is a bit different. I did some googling and several sites were almost verbatim about the benefits / advantages. Here’s one example.
Upshot: yes, it can be expensive to set the fan to “on”, but leaving it that way can improve the temperature consistency, reduce wear and tear from starts and stops, and reduce dust in the house. Setting it to auto means the temperature is controlled mostly in the room with the thermostat (other rooms may vary more), wear and tear with starts/stops of the motor, lower electric bills etc.
Personally with such an old system I wouldn’t bother; its fan is not likely to be high efficiency so it would be expensive, and hell the tenants WANT the central a/c unit to die :D.