'a single company can claim that they have the best safety rating to increase buyers of their cars.'
yes, and it can make the claim without even being the safest car.
The company with perceived to have the safest cars will get the most business, however, I know that i would find it very difficult to tell what he safest car was without road safety statistics from an independent body.
"The action for profit increases the safety of the people driving."
This is simply not the case, the action for profit CAN increase safety standards, IF that is something which customers want, on the other hand if the competing adverts claim they car is 'cool' and 'exciting' I seriously doubt any 'safety' advertising will have much of an impact.
Meanwhile the profit motive, on the same basis, encourages companies to find the cheapest parts they can, and this can in turn reduce safety.
It is only through co-operative behaviour, within an industry, that you can guarantee something like safety standards are placed above profit. (but once everyone is playing on the same level playing field they can still all fairly compete and 'profit driven' motives can operate - taking benefits from both competition and co-operation...)