Why is metallicity important in the death of stars? I always thought that mass was the sole determinant of a star's fate. Then I saw the table here.
So why does metallicity influence a star's ability to become a black hole or neutron star? Does it have as much influence as mass?
I can't give a detailed answer; the details are buried in the depths of numerical stellar evolution models.

The thing that changes most with the metallicity of a newborn star is the radiative opacity of the gas. Higher metallicity leads to more opacity.

This has two immediate effects - it makes energy harder to get out of the stellar interior and makes it more likely that convection will take over.

Convection has the property of mixing up all the material within the convective zone. This can have knock on effects as to how long each nuclear burning phase lasts and how much material is consumed. It also mixes synthesised material from the interior outwards.

A further important effect is that mass loss from massive stars is very extensive and is due to radiatively accelerated winds. For a given luminosity, high metallicity gas is more opaque and easier to accelerate. Hence mass loss is very sensitive to metallicity and determines how massive the star is as it reaches the end of its life. This in turn has a large bearing on what the remnant will be.

There is a further feedback in that the wind metallicity is the metallicity at the surface, but this in turn can be affected by interior mixing that is in turn metallicity-dependent.

If that sounds complicated, that's because it is, and detailed numerical models are required to see how these things play out.