This is "outsourced" to the public sector because it's very hard for an employer to ensure that they're the ones who will benefit from the expenses of on-the-jo

This is "outsourced" to the public sector because it's very hard for an employer to ensure that they're the ones who will benefit from the expenses of on-the-job training - it's hard to take back the training you've given someone if they decide to leave for another company tomorrow. And we have, as a society, decided that we do not want a system in which companies can force their employees to stay. So, public education. In some countries (e.g. Germany) this is actually funded by industries as a bloc contributing to education in their fields, to reduce the free-rider problem while still allowing market signals to operate, but that's still a quasi-public system.