Re: “random slopes should be seen as method of statistical estimation and has no special implications for causal identification in a DAG” - that is how I view the situation with ordinary causal diagrams (usually cDAGs, temporally ordered DAGs with additional mapping from a causal structure, but also SWIGs), when each slope distribution corresponds to a mean-zero prior distribution on the slope and those distributions are all independent, as in classical random-treatment experimental models.
There are however complications that arise when looking at how random-parameter models are recommended and used in practice, which I think need a more general Bayes-net (information network) and hierarchical (multilevel) effect view to understand precisely. That view can map the distribution’s hyperparameters to graph nodes, as seen in hierarchical-model diagrams going back at least to the 1980s (paralleling causal-diagram development). By imposing on those diagrams the same sort of restrictions used to get cDAGs from DAGs, we could interpret them as hierarchical causal diagrams for models that use and extract information on the causes of the targeted effects (slopes).
Consider how we used hierarchical models in Witte JS et al. (2000). Multilevel modeling in epidemiology with GLIMMIX. Epidemiology, 11, 684-688, and Greenland S (2000). When should epidemiologic regressions use random coefficients? Biometrics, 56, 915-921. In those, the nutrients multiply hyperparameters that determine the diet effects (slopes) for a breast-cancer outcome. Using an ordinary cDAG, the nutrients could be graphed as direct causes (parents) of the outcome, with the diet factors as the nutrient parents. The direct diet effects on the outcome correspond to the diet residuals. These residuals are assumed IID mean-zero by the fitted regression model.
Hierarchical graphs may clarify why standard multiple-comparisons (MC) adjustments are often complete contextual and causal nonsense. Applied directly to the diet effects without putting the mediating nutrients in the model, they correspond to treating total diet effects as IID draws from a single distribution centered at zero, which implies complete causal exchangeability for items as diverse as bacon and lettuce. Thus the causal diagram for some MC procedures would have one added exogenous node representing a known constant fixed at zero as a parent of each diet item. Each diet item would also have its own random parent with effect exchangeable with all other random diet effects. This kind of scientifically nonsensical model (which ignores information on known mediators) is also the basis of random-effects meta-analyses (MA) that fail to regress out obvious causes (modifiers) of study results (e.g., bias sources).
Such examples are why I see generalized causal modeling and diagramming as a potent tool for exposing the scientific nonsense behind traditional practices promoted by some statisticians to address MC or MA problems.