Might help to think of it as more of a reprieve. In that the star itself or its remains will still likely end up inside the black holes Schwarzschild radius never to be seen again except as Hawking radiation till the BH evaporates.

That the environment directly outside a BH is energetic enough fling some stuff "up" does not mean the stuff can't/won't fall back "down".

In this case the stuff flung "up" happens to have the necessary properties to trigger star formation further away, which is mostly an incoming shock wave and an preexisting cloud of "cold" stuff.

The stuff the BH is throwing in its shock wave is not going to be "cold". So to facilitate star formation the shock wave has to be less hot / less dense than and maybe slower than a larger BH hole would produce (which would more typically shred the cold cloud to tatters instead of causing it to collapse in on itself precipitating a star)