A mutation question from the Indian National Biology Olympiad
Now, one of them claimed that if the volume were to remain constant,
then mass is directly proportional to density hence the insertion
mutation should cause increase in mass corresponding to a similar
increase in density.

I don't think that's quite how it works.  You don't worry about the volumes being used.  In those experiments (I don't know that anyone does them anymore)  the DNA is in a solution of really salty water.  Larger molecules get pushed more to the bottom of the tube, so the mutant DNA is larger, since it was close to the bottom.
https://embryo.asu.edu/pages/equilibrium-density-gradient-centrifugation-cesium-chloride-solutions-developed-matthew#:~:text=Density%20gradient%20centrifugation%20enables%20scientists,molecules%20based%20on%20density%20alone.

Thus, centrifugal force pushes the larger components of a mixture
farther from the rotor and closer to the bottom of the tube.

Larger DNA means an insertion.  Point mutations are not detectable with this method.