Why this doesn't work like Japan's working holiday visa
In Japan, the working holiday visa is genuinely the main no-degree entry point into paid teaching-adjacent work, because Japan's eikaiwa and tutoring market is comfortable with a mix of visa types. In Korea, the E-2 visa system is far more tightly controlled, and formal hagwon and EPIK contracts are built specifically around E-2 sponsorship, with schools facing real compliance risk if they employ someone without it for classroom teaching.
That means a Korea working holiday visa holder can legally work in the country, but most legitimate hagwons will turn down an application that isn't backed by an E-2 or a plan to get one, because they can't legally put a working holiday visa holder into a teaching role the same way. Some working holiday visa holders do informal or private tutoring, but this sits in a legal grey area worth understanding before relying on it as income.
