So in earlier pages, it seemed like the group was Enthralled, but here they seem perfectly lucid, and the castle mentions a Prize.
Everything we've seen says Mechanisburg genuinely loves their Heterodynes. So the idea of Mechanisburgers willingly harming the Heterodyne seems off, implying that the whole "Chased by mobs" thing is just pagentry.
On the other hand, the threat is real enough that they must Distract the Jaegers and turn off the Castle.
Unless that's part of the point? This ritual is a form of Vote by Potential Assassination. The Heterodyne is tired and probably intoxicated from a night of revelry, they're cut off from the Jaegers and the Castle, they willingly accept the Crown, and then armed townsfolk pursue them through the streets.
If the Heterodyne gets Caught? Maybe they get thrown in the river or forced to wear a silly hat or something, most years.
But, if the Heterodyne has been a tyrant to the townsfolk? If they are unpopular, if they are hated. Well, who knows what could happen.
And the Heterodyne themselves wouldn't know the Mob's intentions, not until it was too late.
Maybe this is Mechanisburg's way of keeping the Heterodynes accountable. Once a year, the Heterodyne throws themselves upon the mercy of the town.
Hated Heterodynes don't survive the Solstice.