We regulate car safety even though there is some competition among car companies. If competition did not lead to net neutrality, then we should still mandate it.

Also, it's true, as others have noted, that competition among ISPs won't be enough to ensure net neutrality because there can't be enough competition (either through regulatory capture or natural monopoly economics).

However, even if there were competition, competing ISPs have the economic incentive to discriminate. The economic modeling is available in the work of Barbara van Schewick (she wrote an amazing, and dense, book on net neutrality). So competition would not necessarily solve the issue, as it does not solve things like car safety (or safe housing). I'm a huge fan of free markets, but there are market failures requiring some government intervention in the form of contract law, property law and its exemptions, and regulation in specific situations.