As far as THE ANDY GRIFFITH SHOW is concerned, if there's one character who really always bothered me, it was Ellie from Season 1, because most episodes with her basically followed the same formula: Andy gets to teasing her about something, she's insulted by his seemingly narrow-minded and sexist teasing, she spends the rest of the episode holding a grudge against him while he thinks she's just being a silly female, another character (like Opie) says or does something that opens his eyes to the error of his ways, they make up and all is forgiven.
Then again, certain characters like that seemed to fall victim to this as well, like the Fun Girls (Daphne and Skippy), for example: you pretty much knew what kind of trouble they were going to get Andy and Barney into as soon as they came into the picture. Malcolm Merriwether (who was actually a decent character) had this problem too.
But Briscoe and Helen, I'm kind of indifferent to them myself. As far as Briscoe is concerned, I mean, Denver Pyle did a great job with that character, and he does have some pretty funny moments (the scene where he keeps yelling at the dinner table comes to mind), but much of the time, yeah, he does come across as another stubborn mountaineer. Helen, on the other hand, I wouldn't say she's a good or a bad character, but she could be rather bland at times, and it's clear the writers weren't even really sure what to do with her sometimes - especially when you consider that, much like Klinger, she was only intended to be a one-shot character, and the writers couldn't even think of a decent name for her, so they just gave her an odd-sounding name like Helen Krump as sort of a joke anyway.