The question being asked is would you allow a character to"give him/herself a Horrible Scar so they could get advantage on intimidation checks?". I (and my players) tend to meet rather terrible gameism with rather dark humored realism.
Assuming the DM was happy with that mechanically, what sort of PC would do that? What sort of person would painfully mutilate themselves to look scarier? There are certainly people that would do it, I would just want to see some consistent RP about it. The person must have some very interesting personality traits.
If the question was "can a character start with horrible scarring for background/RP reasons? can a tribesman of an appropriate culture scar themselves to stand out more? can I brand myself as an act of penitence?" that is quiet different at my table.
Life is not fair to people with horrible scars. The fact is, if a scar is anywhere close to giving you advantage on intimidation checks, most people are going to avoid you. I can't see local socialites inviting them over to tea as easily, fantasy people are not overly PC in my games and nobles are generally not the most caring lot. The person could have got it defending an orphanage, for all the average person on the street cares, it makes no difference.
I got rather sick of some players casually saying things like "I chew off my own thumb to get out of the handcuffs, the cleric can just put it back on later" or "gee I need a disguise, just burn my face off, the cleric can heal it back later". Fortunately, most of my group loathed that sort of thing as much as I did, our current group only do self-harm in epic and important moments.
I have played a disproportional amount of characters with ritual scaring and brands personally and they can be really fun.