There are several questions related to this one but whose answers just raise the doubt I'm going to describe here.
Some facts that are everywhere are
(i) The hierarchy problem results from the fact that the Higgs mass squared receives corrections that are proportional to the square of the next-physics energy scale, if the SM is viewed as an effective field theory;
(ii) This 'quadratic divergence' does not manifest itself if one regularizes the integrals using dimensional regularization. In fact, I have even read in a paper that in dim. reg. quadratically or higher divergences are set identically to zero;
(iii) The hierarchy problem is a real physical problem of the model building and, as such, cannot be 'solved' by a choice of regulator. The hierarchy problem just stop being manifest in some passage of the calculations using dim. reg.
In view of these, my questions:
(1) I simply have no clue about (ii). I have always understood that, regularization procedures being just ways of 'cataloguing' non-sensical expressions to obtain comparable results, the final answer should be independent of them. Is this wrong? How can something so conceptually strong as the degree of divergence of the result be different within the results coming from different regulators?
(2) And, with the mind on (iii), specifically to the Higgs radiation corrections, in which passage of the top fermion loop calculation, for instance, using dimensional regularization, is the 'hierarchy problem lost' (or at least its visibility)?