Isn't the whole advantage that you could get much closer to the center of mass of the black hole compared to a planet, thus gaining much more kinetic energy per unit thrust, but also risking higher tidal forces?
On paper yes, but doing so also reduces the time for the burn. Probes have very low-thrust engines. Even during a jupiter slingshot they barely have time to accellerate much on thier own. Often they do not bother, relying totally upon the grav assist to accellerate. The danger too of a closer approach is that something gets miscalculated. Get too close and an inevitable tiny misalignment will throw you onto a wild unwanted trajectory. There is no gps out there. Knowing exactly where and how a probe is moving isnt easy.
