I think some 1/4 round will clean up a large majority of the eye-sores and will not interfere nor casuse stumbles or trips. The stringer is probably dado/routed out & provides structual support along the wall, removal prior to the new floor was impossible, but still is no excuse for what appears to be a job done while nursing a major hangover. Then again, maybe, he just sucks; the caulk proves it.
After years of practice and when I stated to become a competent and proficient framer, an old-timer gave me all his secrets, letting me know a solid, even and quiet staircase is a testament to a carpenters ability. I knew that advice was to be taken very seriously with no BS.
Sorry this hack did such a crummy job, I’ll post more fix-it hints as I remember them. We’ll clean up all those nosing and top nailed issues. It’s really not all that bad, you’ll see.
-- Ciao, gth.