These same srt files played before or are you saying other korean files played? If these played once fine then that is odd, but not ruling out that you just downloaded improper sub file. Have you viewed the srt content to see if it is right?
You don't state which roku you use. I just visited a family member and two of two movies had srt/subtitle misbehavior. His was a 2XS, and subs would stop and restart.
That's a TCL 4K Roku TV.
I have no ideas but hoped that keeping your thread active someone else may suggest something. Maybe I'll download a korean sub file for one of my movies and see what I get.