An epithet, (I had to look it up to check I was getting this right, btw,) is a descriptive that occurs in place of a commonly used name.
If I know the post anon is talking about, the OP was complaining about people calling John “the short army doctor” or Sherlock “the brilliant consulting detective” or Mycroft “the more cunning Holmes” or similar things within their prose.
The same with many things people flag up as pet peeves, it’s not their use but their abuse that can get people’s knickers in a twist. In balance, they’re fine. Simple epithets can break up the monotony of a passage, and can get you out of the name/pronoun trap (pronouns being a particular issue with two same-gendered protagonists, as you get a lot of the same pronoun going on.)
Mostly, I use them as a tool in POV, and I try not to go overboard. John and Sherlock (whose POV I occupy) are unlikely to refer to each other either vocally or narratively by an epithet unless potentially making an observation through a broader world view. They’re too close for that.
Both, however, are likely to call (or think of) Sally or Lestrade by their job titles almost interchangeably (Sherlock is unlikely to call them by first name at all, sticking to last names or professions. John would probably call Greg “Greg” but possibly stick to Donovan with Sally.)
In the case of Mycroft, I think he would occasional use epithets when considering Sherlock, possibly as a method of distancing himself (that damnable sentiment getting in the way) and is likely to do so to others to maintain an authoritative stance. (In the show, he calls John Doctor Watson when he’s trying to be commanding, and John when he’s trying to patronise or endear for example. That’s not quite an epithet, but it’s an illustrative variation of direct address that could hint at his motivations.)
Basically, I don’t think it’s a case of you should never, ever use epithets. (One post I saw go past said that if you were about to refer to any characters at “The…” anything, then you shouldn’t, and that seems wrong to me) but they should be used with care. It’s very easy for it to become a habit, and for them to slip in unconsciously. For it to become almost a regular thing, which can lead to the writer setting an unconscious rhythm or pronoun, name, epithet use which can read very strangely indeed.
Tl;dr? All things in moderation. As long as you’re consciously aware of your use of epithets and keeping them in line, I don’t think you should worry too much. They’re another tool in your writer’s arsenal, not something that should be shunned as amateurish or distasteful when used in the right place.
That’s my thoughts on it. Thanks for the question, hun!