Grit can be defined as "Perseverance of Effort and Consistency of Interest" [0] so when I see an ad looking for "gritty people" I assume they're trying to find folks who tend to keep working on a problem consistently and won't want to move on until the work is completed.

Kind of a boring response, but I think you're reading too much into this. It's just a word.

[0] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4981570/