Is there any disadvantage or symmetry violation caused by choosing such regularization method? Like, Hard cut-off regularization that violates gauge symmetry in QED. Is there such a practical instance, apart from all the objections based on the problem of rigor(Like the number of Gamma matrices or their representation etc.)?
If not, can one deduce that DR is a "Physically" (and not mathematically!) unbeatable regularization method?
If not, can one deduce that DR is "physically" (and not mathematically!) unbeatable regularisation method?
Why do you say that dimensional regularisation is 'not mathematically [an] unbeatable regularisation method?
The Connes-Kreimer renormalisation theory makes mathematical sense out of dimensional regularisation. The theory is geometric and is related to the cosmic Galois group. In fact, the renormalisation group acting on structures in renormalisation in QFT is a 1-parameter subgroup of the cosmic Galois group.
Well, dimensional regularization has issues with the representation theory for spinors, Clifford algebras, gamma matrices, gamma-five & the Levi-Civita symbol, cf. e.g. this & this Phys.SE posts and links therein.
The most unphysical feature of DR is its infrared behavior for the running of the mass parameter, while the electron mass is infrared fixed in reality!