On reading the section 12.1 of Peskin and Schroeder's (P&S) QFT on "Wilson's Approach to Renormalization Theory", I gathered the impression that in the Wilsonian approach, one starts by analyzing a Lagrangian containing all possible terms compatible with some given symmetry and having mass dimension $\leq 4$. For example, P&S have considered the $\phi^4-$theory and discarded all other terms such as $\phi^6$, $\phi^2(\partial_\mu\phi)^2$ etc in the starting Lagrangian.
However, according to this answer by AccidentalFourierTransform (AFT) and the following comments, in general, P&S's treatment is a pedagogical simplification. The answer says that one starts with a Lagrangian containing all terms compatible with some given symmetry and not just those with mass dimension $\leq 4$. In this scenario, on integrating out high momentum modes, the irrelevant operators contribute insignificantly and effectively disappear from the low-energy effective Lagrangian.
But as I understand, P&S shows the opposite. They show as we integrate high momentum modes, we generate irrelevant terms. P&S's result is also compatible with the fact that Fermi theory is a low-energy theory containing an irrelevant operator. Therefore, we do not "get rid of" irrelevant operators in the low-energy theory. In some sense, we generate them in the low-energy version of the Standard Model.
*How do I reconcile the result of Peskin and Schroeder with the answer of AFT? Following P&S, if we say non-renormalizable operators are "generated" (was not there to start with) in the low energy theory, aren't we saying that non-renormalizable operators become important in the low-energy approximation? But that's wrong.
The answer here by ACuriousMind also says something similar to P&S. This answer also says that non-renormalizable operators are thought to be absent in the original Lagrangian, and they appear when the cut-off is lowered.