We learn from Wilson and Weinberg etc. that there's no holy reason to exclude interactions with negative-mass-dimension coupling while probing physics. However, according to Weinberg, this doesn't mean possibilities can go completely wild, in page 2 of this article(and many other places including his biblicalized textbooks) he stressed:

The second, ‘modern,’ sense in which a theory may be said to be renormalizable is that the infinities from loop graphs are constrained by the symmetries of the bare action in such a way that there is a counterterm available to absorb every infinity. Unlike the Dyson criterion, this condition is absolutely necessary for a theory to make sense perturbatively.

(I should make it clear that here Weinberg is allowing effective theories with infinitely many counterterms)

I simply don't get why this is absolutely necessary. Isn't it quite conceivable that maybe a UV completed theory simply doesn't possess as many symmetries as its low-energy incarnation? That the symmetries of the effective theory are only emergent? With this understanding, even if there are infinities not constrained by the symmetry of the original action, can't we just use counterterms that do not have the symmetry either?