Corporations don't regulate themself. Why would they choose to regulate themself if that means higher cost or less margin or less business opportunities (like selling assault weapons to civilian population) ?
Market can bring prices down if competing corporations in the same sector don't agree to set prices for the same range of services of products. That's it.
Self-regulation _can_ work... if the state is standing there with a big stick willing to use it if the market doesn't regulate itself well.
As a case study, take advertising in the UK.
The Advertising Standards Authority is a Big Deal. It can ban advertising on TV, in print, on billboards and online. Its decisions regularly make the headlines.[0]
It also has no statutory basis, it's a private limited liability company, and they receive no money from government.[1]
In the general sense you're right however I think you're drifting into a philosophical point. Normally it makes more sense to let businesses have a certain amount of freedom to operate and only step in when necessary. The hard part is agreeing where "necessary" is. For example EU tends to have a different perspective to the US on when industries should be regulated.
Market can bring prices down if competing corporations in the same sector don't agree to set prices for the same range of services of products. That's it.
Self-regulation is conservative non-sense.