Yeah, saints are like that, Aqua being a double saint is even more weak against those kind of executions and with the anti-divine spell made by the Amakusa Church inflicts even more damage. More dakka!
Mmmm... since saints have a different nature of their power they can inflict damage to Accelerator, but sure he can add them to his filter if he has the chance.
BT updates:
Volume 7 Chapter 2 Part 5 and 6