Should stepwise regressions also be avoided for exploratory (hypothesis generating) modelling? In a recent paper,  Andrew Tredennick and colleagues (2021) suggested to use the drop1() function in R for exploratory modelling (that is to generate new hypotheses meant to be tested on new data). Their paper is very interesting but, since the drop1() function belongs to stepwise regression approaches (which are badly looked upon), I was wondering whether this procedure could truly be recommended even in this case?
My guess is that hypotheses generated that way will be shaky, at best. But sometimes, shaky hypotheses may perhaps be acceptable starting points to deepen your investigations on a phenomenon, especially when this phenomenon has seldom been investigated and/or data are extremely costly or difficult to collect.
EDIT: I have deleted the second part of my question because I asked it here.
EDIT 2 (Contextual clarification): Ultimately, my goal is to identify the most influencial variables controlling the biological phenomenon I'm studying. All the variables whose effects I want to explore are theoretically important to explain my response variable (they have already been selected based on my knowledge of the study system), but my sample size is simply too small to include them all as covariates in a single "inferential" model.

Exploration means basically, that you can do whatever you want. If there are, among your too many variables, some which are actually true predictors you can hope for the drop functions to find them. You can then gather nuew data to infer, whether these are really actual predictors.
However, gathering new data comes with a cost (or you would have done that before). So the actual question is: Are you ready to gather more data based on nothing more then a stepwise regression approach?
That is not so much a mathematical/statistical question but depends on how costly gathering new data is and how much a positive result would be worth for your further research/career etc.
So, basically, see if you got any better options then stepwise. If not, perform stepwise to reduce the number of candidate predictors. If the result looks really promising, consider to sample new data on those predictors to do inferential statistics on.
