These "savings cards" have a maximum annual benefit applied to them so for those on insurance that do not cover those expensive medications or who are self-paying use up the benefits before year end and do in fact eventually pay full sticker price.
I fit this category perfectly when I was staying in Korea over 10 years ago. I ended up being there on a tourist visa on and off for a few years. I never had any trouble as a white foreigner but did have a hard time making Korean friends, only because my Korean was fairly basic and I didn’t work in an office or go to university. However, everyone I met was very accommodating, whether I was in places frequented by foreigners like Hongdae or Itaewon or in my now partners hometown which is small enough that most Koreans couldn’t place it on a map.
I walked into the government office a couple times and tried to find a path to a legitimate visa and was told each time I couldn’t do it. I found that the longer I was staying, the less comfortable I was with my status. If there was a path for me to be a legitimate resident, I may have stayed longer than the 3 years I was there.