You were advised incorrectly by the CS agent, it is permissible in the event of flight cancellation for you to opt for a flight to any airport BA serves within the same city (eg switching London airport, switching New York airport etc.)

It is always disappointing to hear when an agent does not apply the airline’s policy correctly, because most customers would be forced to take the agent at their word. For a regular FT’er or BA traveller it might be a case of hang up and call again, or insist the agent speaks to their service desk (who would advise them of the correct policy).