@kratzert , I downloaded and checked many stations visually, and I found some strange patterns sometimes:
It seems that either we have stations with years of 0 flow, or the API returns some 0s where it should be missing... It happened only in some basins, so as an advice, maybe it would be interesting to have a look before using it! I do not believe there are such cases of 0s, but trying to be the devils advocate, perhaps they have some very specific hydrology taking place in Korea? Anyway, I found it worth noting for you!

Do you have any ids for which this happened? Would maybe facilitate the research

Ow, silly me, for sure it is needed the ids. Sorry Freddy. Here it goes some examples (for streamflow): 1018680 (from 2010 onwards), 1001605 (2013/2014), 1001607 (2015 onwards). For stage I did not get any issue.