This is indeed a brilliant article worth multiple rereads. For the linear-Gaussian case that @Stephen wisely focuses on to avoid the whole non-collapsibility hoopla, I think we can distinguish (dichotomize 
) between the design-based versus the model-based approaches, whereby model-based approaches use a superpopulation to provide aleatory probability interpretations.
In Senn’s terms, the randomization distribution (RD) approach is purely design-based but less practical. The multivariate (MV) and linear model (LM) approaches both effectively invoke a superpopulation. But the purely LM view treats the baseline prognostic covariates as fixed and does not justify omitting them. The only random object is the outcome. Conversely, the MV view treats also the baseline prognostic covariates as random and jointly distributed with the outcome Y. This is the old Galton–(Karl) Pearson correlation worldview where everything is jointly normal (as recounted in the Aldrich 2005 paper cited by Senn) and the marginal of a multivariate normal remains a perfectly well-specified normal. Thus, in the MV view, conditioning on a subset of random baseline prognostic covariates is legitimate. That is what gives conceptual permission not to measure everything. LM is the worldview under which omitting an observed prognostic covariate is outright misspecification (the very discomfort that drives Senn to MV).
As an aside, RD is an example of frequentism-as-model (FAM) where the repetition (frequency) is not derived from a superpopulation but re-randomization (a real and performable operation). Thus, RD is the one framework where the frequentist probability nearly exists qua design. As also noted here, particularly by @f2harrell, one may instead dispense with the superpopulation clause by putting epistemic probabilities on the parameters via the Epistemic-Probability-as-Model (EPaM) view.