I would treat it the other way round. Inspiration should not be something that you are desperately holding onto in case you need it. Better if it refreshes once per long rest. And that way, it is not a game balance issue how often the destinies trigger inspiration.
And a significant problem that I have seen in play with some of the other destinies is even remembering to get inspiration at all, then needing to ask and prod the DM as to whether you get it in given circumstances.
As an example, I have a Warlock with the Coming of Age destiny. I have a tendency to sit on the inspiration to use it when I need proficiency or say to reroll a saving throw. As a result, I seldom use it. The character's triggering criteria to gain inspiration ("when you achieve a personal milestone; join a new guild or organization, travel somewhere new and far from home; accept a new major quest or mission; change worldviews and grow as a person") and not using inspiration often, mean that I have used inspiration a total of 3 times in 9 levels. That makes inspiration an unimportant mech complication that is an afterthought in play. If it was gained once per long rest, or once per session, I would use it more often and it would be a meaningful mechanic.