The issue is not "plot immunity" - it is power. Your typical horror protagonist doesn't have any. He or she is a shlub without the skills necessary to take on the darkness. This does not describe most fantasy characters. There are many kinds of fear. The fear in horror is a helpless fear, the fear in fantasy is not. In typical horror the dangers are strange, outside the protagonists' experience. Fantasy heroes are specialists in the strange, dangerous, and monstrous.
All that THAT means is that the typical horror protagonist is 1st level...and may not be by the end of the plot arc (depending upon whether its "Everybody Dies" horror or "Heroism Prevails" horror).
And as Remathilis points out, sometimes the horror protagonist is actually quite experienced, just not as dangerous in one-on-one combat as his foe...meaning he must use skill, wits and occasionally allies to defeat his foes.
Take a typical zombie story, for example. A given zombie isn't appreciably more powerful than any protagonist (and in some storylines, are
less powerful)...its just harder to make them go down and they never tire. The protagonists, OTOH, do tire, so the horror in zombie fiction is, in a sense, about attrition and inevitability...being overwhelmed by by a tsunami of undead that, though you can resist it piecemeal, you cannot stop. With skill, wits and allies, however, you may live a long, if grinding, life.
All that said, however, I'm not arguing that D&D is a horror game. D&D is, at its core, more fantasy than anything else. But even so, its nature is such that it may easily venture deep into the territory it shares with horror and even sci-fi. And in some hands, in select campaigns or settings, may even resemble one of those other genres more than its core.