I wrote a huge Wall of Text™ in a forum post about Fallout: Equestria and Fallout Equestria: Project Horizons and I might as well as post it here as it sums up why I like the fics a lot (Warning: some fairly minor spoilers here, original had a few major ones but they have been edited out as there is no hiding of spoilers here) - tl;dr version: Me probably over-analyzing a fanfic like the pretentious jackass I am.
Meh. I'll admit that F:OE could have probably been cut by a lot - but a lot of those extra bits are fun! I do think you may be missing the point of the fic though.
While there is a lot of grimdark, you also see occasional bits of whimsy and good as you go along in the story. Mostly these good bits are seen in a few of the memory orbs showing pre-war Equestria and some bits in the present which show hope for the future. A large part of the story is about how Equestria and the ponies eventually fell little by little (often through good intentions) into a horrible uncaring and hateful society. The other large part is finding the good that is deep down and going on to try and help make things better despite mistakes because giving up to wallow in one's self solves nothing and can in fact make the situation worse. One has to dare to care, so to speak, taking others' feelings into consideration, and doing what is right even if it is unpopular. Fallout seems to be kind of used as a metaphorical window dressing representing the utter break down of society to tell the true meat of the tale and it diverges heavily from Fallout's feel because of its themes (and also because the world of Equestria has a lot of fantasy in its roots since Faust used LotR, Narnia, mythology, and a few fantasy novels in originally creating the setting. One can't really get rid of them with out tossing out the core of Equestria as well).
I think it caught on because it is kind of a zeitgeist that mirrored how a good number of Bronies felt; the world is uncaring and in a metaphorical shambles (what with the economic situation and exploding political tensions) and this fic presented a character who rises above the ever-present sense of anomie (represented by the wasteland) and is inspired by visions of the old pre-conflict Equestria (our ponies) to try to restore the good using the Mane Six and pre-corruption Equestria as a model (for some reason a song is coming to mind here :p). Along the way LittlePip sees how good intentions and misunderstandings led to the corruption and destruction of old Equestrian society and learns from them and he also learns that even if one makes mistakes and fails causing hurt then one shouldn't give up but instead one should try harder to not make the mistake again in the future. Additionally, one should try to make up for the damage by doing good and continue by concentrating on the good that is inside us all.
As for the side fics: I've only really read Project Horizons which has similar themes but tends to have a more limited and personal scope. All of the mane characters have deep psychological wounds and the core focus is not on the individual vs. the sense of anomie in society (although that is still a major theme), the focus instead is more on dealing with one's own personal trauma and emotions (punctuated by light-hearted moments of awesome comic relief like the 'Queen Whiskey' bit). One deals with the emotional storms by finding strength in oneself through knowing one's self and the support of others. Another major theme is the dangers of repressing hurt, pain, and bad emotions until they explode and cause harm. On a certain level, I kind of find reading it a little therapeutic and kind of identify with Blackjack - although I've experienced nowhere near the sheer level of trauma she goes through - Blackjack is often the 'strong one' in public for others (heck, a former roommate openly said I was the person who most 'had their shit together' and I've ended up playing therapist to a bunch of people) but deep down inside Blackjack has a fair share of repressed guilt and trauma. Repressing her guilt, emotions, and trauma nearly destroys Blackjack (several times) and it is only when she starts to know herself and accept the guilt, emotions, and trauma that she can begin to heal with the help of her friends.