Congratulations on a successful fledge! And welcome to the forum.
Yes, you should definitely clean out the old nest. Wear gloves and put the old nest in a baggie and dispose of it. Leaving old nest material on the ground nearby can motivate predators to search an entire area for the source.
Bluebirds and other cavity nesters will build a new nest on top of the old one if you leave it. The resulting height of two nests puts it dangerously close to the entry, providing predators an easier reach inside.
If you apply grease to the pole make sure it’s under a baffle so the birds can’t contact it.
Good luck and let us know if they nest again.