In my opinion a significant source of the rumble on non-moulded wax ones also are / were the lathe on which a given cylinder was cut or dubbed.  Add to that, the aberrations (that develop as time passes)  in concentricity from wax & celluloid moulded ones  & you've got considerable low frequency crap that ordinarily would not be detected by  acoustical reproducers.  Filtering the signal is thus often warranted.

Mechanically adjusting for improved concentricity is often not as simple as one would hope.  The physical distortions of the tubes are not uniform: a setting for one end is not necessarily optimal for the center or other end.  Theoretically, adjustments for portions of the cylinders should be made individually (in whatever increments you deem desireable: every 1/2", 1" etc of grooves) & the file then assemble edited.  I do this comparably for disks when changing the arm's tangent (for disks  with extremely disparate radii of inner and outer most grooves).

Another effective technique for dealing with cylinders' tubular eccentricites is to run them at 1/2 speed and then double the pitch for playback.

I don't know if anyone's undertaken to make for themselves, or to offer them to others, but it'd be smart to have a machinist precisely (accurate to .0001")  mill a suitably configured cylinder that'd be analagous to a test record or alignment tape.  It'd be made out of something soft that would retain its shape but not jeopardize styli & their cantelevers, i.e. teflon.  It wouldn't necessarily have to be grooved, but would provide a baseline of measurement determining how much rumble's generated from any playback mechanism versus that emnating from a particular record.

Also, it's important to not assume that any given groove is "vertical".  As with all mechanical devices, tolerances are involved.  IF a groove induces a stylus to meaningfully oscillate at some angle other than true perpendicular (+ or - the theoretical standard), then unnecesary low and high frequency noises will occur. This is remedied by continually adjusting the phase relationship of the stereo cartridge being used to track the grooves.  I call the process "vectoring". It's extremely beneficial for ALL monaural grooves.