With all else being equal (which it almost never is), core size can definitely have an impact on performance. Good luck getting an actual core weight sans all the casing, potting, etc.
EAR MC-3 and MC-4 add weighting to the box so stiff audiophile cables don’t wrestle it around - which is really nice. Almost all the CineMag MC variants use the same size can - but interestingly Quadratic (MC-1) has their CineMags custom made with at least double the usual height (presumably with a proportionally larger core). It featured stronger than usual bass response, along with a slightly relaxed treble (this behavior was consistent across MM stages and cartridges).
Lundahls (9206, 9226, 1931) use amorphous cobalt cores, rather than laminations, which must weigh differently - and I also wonder if this core material explains my aversion to them.