I’m worried my long period on student status could hurt my employment-based case

Many people have maintained status for a long time through a language institute. Someone on student status generally won’t have work authorization, so USCIS will certainly ask how you supported yourself and covered living expenses for so long without it. You should prepare in advance to document how you funded tuition, living expenses, and — if you have children — childcare costs, such as records of money transferred to you from Korea.

If you worked on campus under a school-issued SSN, you may have some documented income, and you may also have wages earned during OPT. Since USCIS may otherwise assume you supported yourself through unauthorized work — unauthorized work = an immigration violation = green card denial — it’s worth preparing carefully to avoid that chain of reasoning.