My mother in Daejeon goes for regular thyroid check ups every few months and usually books in advance at a local clinic, where her wait is about 10 to 20 minutes. She prefers morning visits since afternoons can be a bit busier. She once went to a larger hospital for additional tests and ended up waiting nearly 2 hours even with an appointment. Her simple advice is to use clinics for routine care and reserve big hospitals for specialist needs.
For non emergency medical visits in Korea, wait times are usually quite short often around 5 to 15 minutes if you have an appointment. Even walk-ins tend to move fairly quickly compared to many other countries. From my experience, using online booking makes things much smoother. I recommend booking in advance, arriving a bit early, and bringing any relevant medical information even a translated list if needed to ensure a hassle free visit.