The neutron star you refer to is due to a type Ia supernova, this event produces neutron stars with masses about 1.4 solar mass.
If you replaced the Sun with a neutron star of the same mass (1Mo) then there would be absolutely no changed in the orbits. If you replaced it with any stellar object with a different mass then the orbits will change.
The point you made about surface gravity is true, but it's due to the mass being compressed into a smaller volume, so the radius is smaller.
By studying this equation it's clear why this is true.
So, when we replace the neutron star as the center of the Solar System, the planets will be drawn by the gravitational force of a very large neutron star. As a result, the planets are also destroyed and the material is drawn toward the neutron star. Could be, splinter planets in the Solar System destroyed by the pull of this neutron star will form the accretion disk around the star.
So this statement is a bit misleading, only if the mass of the neutron star was larger than the sun this statement is kind of true. Neutron stars can have masses as low as 0.5 solar mass.
It's cool dude, just trying to clear up an important point about the physics. I have the Bsc in Astrophysics so I have to use my knowledge somewhere :)
I hope this information helped. :)