This is my view and not necessarily that of the AA.
Business profit is made in the risk assessment of how many people actually break down compared to how many paid for the service (like all insurance based products) It costs to send a patrol/garage same as it costs to recover (p/per mile) which is not charged to the customer. I think the biggest profit is in cross selling other products.
Many years ago roadside service (RSS) was subsidised by other profit areas within the business. RSS was loss making; hence diversification into maps travel and insurance etc, although RSS was the main reason people joined the AA.
I am sure that some profit is made from selling parts at the roadside but paying commission to the patrols will offset this. Incidentally parts turnover is fairly low in my van; if you remember that list from Manchester the top 10 breakdowns do not always need parts. I usually have to locate the part and the customer pays for it direct...and they donít work 24/7.
So to answer you query, profit would be made if less people broke down, the more that break down the more it costs = less profit.