In the event that the INSERT triggers a constraint that the UPDATE fails to resolve, it will still fail in exactly the same way that running the ON CONFLICT on a specific constraint would fail, so it's not like you gain any extra value from specifying the constraint, is it?
I don't know why I wrote this paragraph, it's just the product of me thinking of something else at the same time:
UPDATE obviously doesn't resolve a conflict as such.
Thinking about it more, I suppose if multiple constraints end up triggering for the same INSERT, it would require UPDATEs of multiple rows. Is that the issue?