I'm far from an expert in Japanese and what worried me about my understanding of this description is that it doesn't seem to fit the character it's describing.
That’s alright; there seems to be quite a bit left unsaid in this description and I had to fill in the details because a literal treatment would be nearly unreadable (and very contorted in English besides).
This is from Brigandine Grand Edition and most of the Rune Knight descriptions describe consistent characters that match their portraits. Since I'm not an expert, I thought I owed it to the fans to make sure I wasn't bungling this one. This knight's (Vanessa) portrait shows her in full plate armor and looking rather serious; it doesn't seem to match a "bar maid" who's so oblivions that they didn't notice they had Rune power.
The implication I kind of sensed (without hard evidence) was more that she was willfully ignoring business that might get someone killed if they poked their nose in it; it’s not necessarily about incompetence. Probably quite the opposite (fiction is full of “humble tailors” and “innkeepers” who may or may not still have a foot in the game...). She might have even been working there, waiting to get revenge! It’s not said nor implied here, so I don’t know.
See, there is a bit of tense trouble sometimes: いていた is fairly close to English’s past perfect tense in intent. One could more accurately (but confusingly) say she had been working
at (or owned?) a bar/pub some time after—probably not long after—retirement (and felt that if she’s clearly back in business by the time you read the text, a simple past tense will work fine and sound less awkward).
The phrase 〜を機に (___ as an opportunity) would usually be translated “take advantage of ___” or something like that, but it’s not really meant to sound that terrible, I don’t think. Rather, Dad dies, and suddenly there’s maybe like no duty to live (and die) by the sword anymore. Or something like that. Who knows? It’s (again) not said here, and I get the feeling it was intentionally left unsaid. This could have also been translated “took leave to” if I wanted to take more liberties and if the whole story supports it. Sometimes people intend to “take leave” permanently, without permission, you see....
And the line about her father seemed like it was out of place- stuck in the middle. I also couldn't find the exact form of the word that you translated as "vulgar" in my references. Is that something of a slang phrase?
You may find it under a heading of 柄の悪い or 柄が悪い (in my dictionary, it is under 柄 alone). I’m not sure I can accurately explain the mechanisms and situations where の, が, and は can be exchanged for each other, but trust me, this is one of them; in this case, the は is contrasting a character flaw with a character strength. Which was another difficult translation choice: 筋の通らない means “unreasonable”, “illogical”, “unjust”, and “immoral”, sometimes all at the same time. And I think it does here. She is unpleasant in some way, but what she truly hates is things, any things, that are just plain wrong—this would not surprise me in a soldier!
“Vulgar” is only one of many possible translations for 柄の悪い, but I thought it the safest fit because she’s probably coarse, rather than bad. Some things are better left for the players to disambiguate on their own anyway (show, don’t tell). Other definitions of 柄 include a quality, a right, a standing, or one’s physical build, so there are many parallels to Western concepts of having good or bad character that one can use to translate this term, if it turns out that’s not an appropriate fit.
I get how ルーンの証 would likely be "Rune seal", but in this case, I think it is "evidence"- the game premise is that rare individuals manifest the power of Rune magic which lets them control creatures and cast spells etc. I think this is saying she wasn't aware she has the "runic gift". There is probably a dialog event when this character is introduced that explains the scenario better. I'll make a note to check context clues for the exact meaning when I get to that section of the text.
I left it untranslated since it strikes me as something that needs deliberation and the rest of the context.
Aaaaand because I couldn’t finger which game it was based on that alone anyway