Yeah, the "vestige abilities are supernatural" applies to all vestige abilities, not just the 5-round ones, and explicitly includes ones that mimic spells. Even to ones that look extraordinary and are always on, like Sneak Attack. Having a vestige ability usable a certain number of times per day is unusual, but also not unprecedented: See also Tenebrous's Flicker.

While it's always possible for something to make a specific exception, that general rule is broad enough that I'd say that it applies unless something very explicitly says that it doesn't (i.e., "This ability is a spell, not a supernatural ability", not just "cast").