TGIQ seems to be invested in shifting POV, showing how the myopic fatalism of Angel's LA is not THE truth, so to speak, and that in Sunnydale or in Rome, Angel's a very different looking ~hero. I think the contrast is interesting because of how the Roman absurdity interacts with the LA direness of the Wes/Illyria scenes (men chasing ciphers of women they ~love), and this tonal dichotomy is again reflected in how Rome-TGIQ contrasts NFA.
However, it seems like Angel's character flaws being given center stage for comedic purposes and demonstrating POV distortion to undercut the serious angst drama of AtS isn't what most people want out of this story. I read a lot of critique about how this episode messes with the serious tone of the final episodes and I think -- yeah, it really does, that's why this story exists in the first place.
It feels like a majority of fans really don't want that dire tone messed with. We can have a serious undercutting of Angel's heroism, but a humorous undercutting is beyond the pale because it's too serious business now and he's a ~hero~ without question (except for the fact that the entire series is largely interested in questioning Angel's heroism so).
I've always felt that "Smile Time" and "The Girl in Question" are such extreme farce in direct reaction to how seriously Angel's taken himself over the years, and this seriousness gets amplified to the extreme in Season 5 (especially without Cordy there to slap him upside the head and say "snap out of it" like in S1/2). Angel can only take a vacation from his dour, dark drive if he actually leaves the country. That's how fatalistic he's become.
Even the OTTness can be explained in-story as the stark relief one seeks in order to escape. Angel's running around Rome so desperate to escape the W&H noose, hoping to rediscover a simpler purpose like saving the damsel, is an attempt to return to his basic definition of heroism (chivalry, even). After a stultifying season of gray morality, Angel wants black-and-white goodness to take his ease. (Which is also why I buy Angel's characterization in "Chosen" -- simpler times looking mighty fine, oh yeah.)
I guess I find it all too realistic for people to act ridiculous in reaction to unrelenting psychological pressure. This is Angel's snapping point, his last desperate attempt to be simply good before he buckles down and determines to do evil in the name of good. He desperately tries to make his life simple, to have it all make sense in his own terms of how life works (he's a hero who saves women and defeats evil "I kill the bad guys"), and Rome shows him up on how THIS DOESN'T WORK in a symbolic microcosm of his current situation in LA -- he's trapped, running in circles, all because he did what Lindsey warned him not to do: don't play by their rules.
Yet the tonal shift stalls the momentum of the rising action leading up to the epic finale, right? Well, what if the point is a deliberate stall?
Angel wasn't ready to fully sink into doing evil in the name of good, so he runs away for a weekend bender where he's gonna save Buffy from a morally gray figure. It's a classic denial of his darkness and it doesn't work. One can even read the shadow figure of the Immortal as symbolic of Angel's darkness keeping him from being a white knight; his shadow figure foils Angel's attempts to be ~pure~. As Angel's running around attempting to save Buffy through brute force, his shadow figure Immortal is manipulating his enemies into defeating themselves (how very Angelus of the Immortal, right?). Having failed to be redeemed once again by becoming Buffy's hero (by a manifestation of his own symbolic darkness, no less! There's a reason we never see the Immortal -- Angel is his own enemy), he returns to LA and continues with his dark plan instead.
Just as the scenes are split between dire angst & satirical farce, Angel is also attempting to split himself. Angel is running away from his darkness, trying to recapture a pure light, and instead his darkness is ready and waiting to conquer him in Rome; his darkness wins the day, sending Angel back to LA with his not-so-white-knight tail between his legs. This episode is about dichotomy and forced reintegration (see Illyria pretending to be Fred, see Angel pretending to be a white knight -- they're mirrored here which is fitting since Illyria is the one who gives Angel the answer in "Time Bomb").
Angel's trying to be the white knight one last time. He's inhaling deep of the pure heroism he aspires to before diving into the darkness his plan demands of him. Frankly, I think it makes him more interesting and it shows him in a good light that he wants to be good and that he hesitates -- and even psychologically balks -- to do evil in the name of good.
Ultimately frustrated by his attempt to escape, Angel continues on with his typical m.o. because this is how he fundamentally functions -- kill bad guys. All the while his mission of helping the helpless is collecting dust on the sidelines as his connection to the helpless is remote and out of reach: he no longer gets visions of victims in need, and Gunn is the one who connects with Anne; Angel can't connect with other people, instead his challenge is to reconnect with the symbol of his own displaced humanity: Connor. Instead of helping the helpless, Angel's actions instigate an apocalypse raining down on his city and sending it to hell. So, he failed with Acathla, but he rights his wrong in NFA. Everyone's going to hell; no worries, Angel saved you a seat.
TGIQ is supporting evidence of the dark underpinnings of NFA. It's comedy and it's POV distortion serve as a counterbalance for the unrelenting darkness of NFA. The distortion is meant to be jarring, to make you question, to illuminate how one of these things is not like the other. Unfortunately, the jarring tone is mostly dismissed as a mistake and a poor choice by fans, commonly citing the reason for TGIQ's wrongness as its failure to fall in sync with the dark mood of NFA.
...which is rather the point. Humor don't live here anymore. We can't laugh off our pain, we can't heal (forever cursed!), we can't be redeemed (bye shanshu!), so we must kill it with fire! DIE EVERYONE DIE.
(And if anyone doesn't already know I read this darkly, where ya been, right? And really how else are we meant to read noir but as dark? It's in the name! :P)