What in-game event makes the Undertale Pacifist Route possible?

Solution 1:

As an initial matter, Toriel is the only character whose actions demand explanation. Papyrus shows up because Flowey told him to, and everyone else shows up because Papyrus told them to. We know this from an exchange between Alphys and Papyrus, right before Flowey appears and incapacitates everyone.

Unfortunately, that's where we run out of evidence. As far as I can tell, the game offers no explanation whatsoever for Toriel's presence in the Pacifist Route but not in the Neutral Route. In the absence of canon evidence, I would speculate that one or more of the following happen differently (in roughly decreasing order of likelihood):

1. In the Neutral Route, Sans stops Toriel at the Last Corridor, probably because he doesn't recognize her and/or is concerned for her safety (there's a human and a boss monster fighting just a few rooms away!). They recognize each others' voices and proceed to have a lengthy conversation. In the True Pacifist Route, Sans does not appear in the Last Corridor, so he wouldn't be there to stop Toriel.
2. In the True Pacifist Route, Flowey approaches her in the Ruins and guilt trips her into following Frisk. In the Neutral Route, he does not. But it seems odd that she never mentions Flowey after the final boss, so I'm not sure this makes sense.
3. As you suggest, Toriel was following Frisk the whole time. But I'm skeptical of this, because in the Neutral Route, you can backtrack all the way to Snowdin at any time, and you never encounter her. By contrast, Flowey does occasionally appear just out of sight behind the player, and quickly burrows underground should the player backtrack.
4. Papyrus told Alphys, Alphys told Mettaton (who does very briefly appear in the scene before the final boss), Mettaton told Napstablook (his cousin), and Napstablook told Toriel. We know Napstablook is able to enter the Ruins, but it's unclear why they would approach Toriel in particular, so I think this is pretty far-fetched (especially since Napstablook seems totally unaware of the entire situation during the epilogue).