See this post at A Fine Theorem and the discussion in the comments: link.
Structural modeling and identification give rise to lots of possible combinations. Randomization (and its analogues) can non-parametrically identify ATEs or LATEs and other things that can be constructed using only marginal potential outcome distributions. But as, e.g., Heckman et al. (1997; link) have shown, there are pretty strict limits to what randomization can do to identify parameters from the joint counterfactual distribution. Behavioral assumptions, the basis of structural models, “fill in” the information needed to proceed with estimation tasks that require more than just the marginal potential outcome distributions. Along similar lines, Chetty (2009; link) has shown how behavioral assumptions can motivate the interpretation of non-parametrically identified parameters as “sufficient statistics” to judge welfare effects (or, at least, to put bounds on such effects). The general principle behind all these combinations is that models (“structure”) fill in for what randomization cannot identify non-parametrically (that is, “on its own”). An issue in the discussion linked above (in the comments especially) is whether and when it is okay to just work with what is non-parametrically identified.
Perhaps a key source of the tension in “randomistas versus structuralists” debates is a difference in opinion over where we should draw the line between acceptable and unacceptable use of structure to “fill in.” Even randomista papers sometimes apply bits of structure to decompose (L)ATEs to link results to theoretical claims about behavioral mechanisms. Here is a very barebones example from Duflo and Saez (2003): link. So the debates are not black versus white. There is probably less controversy over the suggestion that we shouldn’t use structure to identify parameters that could in principle be identified with an experiment or natural experiment. E.g., introducing structure merely to identify a LATE (selection models, anyone…) probably rub a lot of people on both sides of the “debate” the wrong way these days. (And even this would be seen as a step above completely hand-wavvy identification strategies like plopping an ad hoc array of covariates into a regression or matching algorithm…)