As seen in Einstein's 1920 address from the University of Leiden, for example, he did consider it meaningful to distinguish between the presence and absence of the aether, and certainly he seemed to lean toward, rather than deny, its existence -- albeit an inherently four-dimensional aether without motion, semantically no more than a placeholder for "what spacetime is".
At the same time, as we can read in this section on his geometric approaches (second paragraph), his later mathematical models
eventually centered around treating both the metric tensor and the affine connection as fundamental fields.
Of course, his belief in the aether, as far as it went, could be independent of the mathematical details. Nevertheless, I'd like to know if he ever gave any indication that they influenced each other in his thinking.
In particular, when Einstein admitted torsion and/or non-metricity into his pseudo-Riemannian connection, did he suggest either publicly or privately what these deviations from the Levi-Civita connection meant to him, in his conception of the unified continuum? Like, did he imagine the torsion to imply that the aether is twisting, and non-metricity to imply that it is stretching, in some real sense, akin to if not identical to an elastic medium? If so, how far did he consider it possible to take the analogy? And did any such analogy inform his notions of what the governing equations of the aether-continuum should be? For example, did he say how he suspected torsion and curvature ought to interact with each other qualitatively? Conversely, when he contemplated a candidate field equation arrived at via considerations of symmetry and beauty, would he try to translate it into a description of the behavior of the continuum?
(Note that I'm not asking anything about his address linked above, as he mentions only the metric and not the connection there -- I just included it to show his inclination toward the aether. I'm asking if there are other references that show his thoughts on the physical meaning of the connection.)