Basically that's Quality Assurance. While Lambda School might be doing good by having the right incentives for all involved parties, a lot of money could be made by tricking people to enroll into questionable schools. So it's for the public good to regulate that market.

The question is rather why Lamda School didn't get themselves approved?

There have been plenty of instances of private, for-profit schools that operate in a predatory manner. States have a legitimate interest in regulating potentially predatory behavior. So-called code bootcamps are not special in this regard.

Private schools that don't operate in the student's best interests pretty quickly can be a larger economic problem for everyone. These schools can be expensive and having a bunch of folks who want to do the work to move up getting ripped off is not in society's best interest.

Same reason you can't just go out and buy a car with your own damn money and legally drive it without a license. It's in the public interest that certain groups meet certain requirements in order to perform certain tasks. There's plenty of problems that come out of this, but generally people who support it think it's better than no regulation at all.