I saw a photograph online recently that showed many upward electrical discharge leaders from structures trying to connect with the discharge coming down from the cloud. Very cool photo.
The air terminals only attempt to guide the high current discharge around the structure. They do appear to have an effect at preventing or reducing damage.
In Franklin’s day, the fire insurance payouts were reported to have decreased after church steeples were outfitted with earthed air terminals. It may have been just the presence of the earthing wire at the tip of the steeple and not the air terminal itself that made the difference.
In chemical plants and refineries, we go to great efforts to ensure that the steel structures and equipment are adequately bonded to earth through the foundations and through connected earthing rods or grids. This is an attempt to ensure the the ground potential and the potential at the tip of the structure are the same - as if there was no structure there. It also ensures a path to earth that can handle high currents. However, tall structures, poking up into the atmosphere can distort and compress the shape of an electric field forming above the earth due to a storm cloud in the area possibly increasing the chance of a hit. Not that a cloud discharge will hit there every time, but higher odds though.
Just look at the lightning protection towers in place around rocket launch facilities in Florida used to reduce the odds of a hit to the rocket.
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