For clarification: When the original card was trashed, would it or would it not have the duration effect on the next turn?
Trashing a Duration card doesn't stop it from functioning (except for below-the-line stuff e.g. Bridge Troll's cost reduction).
But it should stop it from functioning - if there's no card in play, how can one be sure that the players will remember all the effects on that player's next turn? If this Bonfire-a-Duration scenario ever came up in my group (and why on earth would it?? ) I'd certainly say that it means there's no duration effect on the following turn.
To me, that's the key issue in these kinds of discussions. Duration effects must be represented by cards in play; one Duration represented by the card itself, two by Throne Room or whatever caused a second copy of the effect. And in the case of Bonfire, the number of Duration effects on the following turn must be zero because there's no way to have any cards in play to remind everyone. (Again, not that there's any reason I can imagine why anyone would Bonfire a Duration...).
Using this interpretation of trashing Durations would allow the errata on Procession to be lifted. Go ahead, Procession a Caravan, you get two cantrip effects and draw no extra cards at the start of your next turn (that's something I could see occasionally being worth doing, at least).
But that's based on the usual principle that an action can still continue to be played from the trash. E.g., if you Throne Room a Tragic Hero, and it gets trashed on its first play, you can still get the +3 cards +1 buy
and the treasure on the second play
And this is precisely
why Procession got its errata - because without that errata you could do exactly that. But in practice, it could be easy to forget that you'd processioned a duration on your previous turn. So, to prevent that confusion, the non-duration qualification was added to Procession
EDIT: I just realized I misread the second part of your post. It's certainly true that if it worked that way, we wouldn't need the errata, but that would be an actual change in the rules, not just a minor difference in interpretation - you'd be adding a rule that stops Duration cards, and only Duration cards, from working when trashed, as opposed to changing the way one particular card (Procession) works to avoid a potentially confusing effect of an existing rule