Why does stellar rotation not shred the Chandrasekhar limit? Investigations in the 60'ies about the question how the Chandrasekhar limit is modified by allowing a self-gravitating mass of cold fermions to rotate seem to have come to the conclusion that the limiting mass is only increased by percents or at most a few tens of percents.
I want to confront this conclusion with the multitude of rotating pressureless self-gravitating disk solutions (to Euler or Vlasov-Poisson equation). The mass of these solutions is not bounded above and -wonder oh wonder!- we do not even need any pressure to counteract the gravitation pull. Rotation suffices to do the job. Granted, none of these solutions seem to be dynamically stable, even to linear perturbations. But already since Jacobi opened his mind to his famous ellipsoids we've learned not to constrain ourselves to strictly stationary configurations.
This leads me to ask two questions:
Q1: Where -in the theoretical arena- does rotation fail to significantly increase the Chandrasekhar limit?
Q2: Can I not see a typical spiral galaxy (e.g. our own) as a single pressureless supermassive star, its flagrant violation of Chandrasekhar's bound due to rotation?
 A: I am not sure that your information is up to date. Indeed, papers by the likes of Anand (1965) do suggest that rapid rotation can only increase the Chandrasekhar mass for white dwarfs by a few per cent. However, starting with Ostriker & Bodenheimer (1968) it was realised that this only applies to uniform rotation, where angular momentum is transported highly efficiently. If the star is allowed to rotate differentially, then the Chandrasekhar mass can be increased by factors of a few.
The physical reason is that in a uniformly rotating body the ratio of rotational kinetic energy to the magnitude of the gravitational potential energy cannot become very large before mass-shedding at the equator occurs. ($T/|\Omega| \leq 0.007$ for spheroidal models, according to Shapiro & Teukolsky, Black holes, white dwarfs and neutron stars, p.177). However, differentially rotating models do not have the same constraint and Shapiro & Teukolsky find a maximum mass for a secularly stable, differentially rotating configuration (with $T/|\Omega|=0.14$) of 1.7 times the non=-rotating Chandrasekhar mass.
So I guess this is the root of a solution to your question. A spiral galaxy does not rotate uniformly.
