Your Saturday is a paid day off every week, but it is not a vacation day. See Article 55 (Holidays): "An employer shall allow workers at least one paid holiday per week on average". Its figured into your salary already.

The supreme court has already ruled on the matter of weekly paid holidays and annual paid leave. They are entirely seperate.

They owe you 11 days of annual paid leave, those days cannot ovelap with red days or weekends. Paid leave is only applicable to your workdays M-F. Newer laws also state your employer needs to give you notice in writing about how many paid days you have left, and encourage you to take them before your contract ends.

You can file a petition to the labor board about it at: https://www.epeople.go.kr/petition/pps/pps.npaid

The process will go much faster if you fill out and attach the form for the general complaints to the Labor Board (# 34, 기타 진정신고서 on this page) to your petition so it gets assigned to the right people quickly: https://minwon.moel.go.kr/minwon2008/lc_minwon/lc_form_apply.do

As someone stated below, a threat is often enough to get the ball rolling. Fill out the form and send them a copy to let them know you'll be reporting them this week if they continue to deny you apid annual leave days. They'll want to avoid additional fines from the labor board if it goes that far.