Yes, SR is still a classical theory and there is an objective reality behind it. My point was that the "mystery" in the twin "paradox" case was due to a failure of the thinker to wholly accept the relativity in the theory. QM is likewise a relativization, (word?) it however relativizes the objective state (in CI), that being relative to a choice of compatible observables. It is fundamentally different from classical theories, including SR so you won't find the same kind of loss of objective reality.
As to consensus among physicists CI is also referred to as the Orthodox interpretation. It still is the leading view:
https://arxiv.org/abs/1301.1069
The "sexier" interpretations (EMW, BPW) get over-represented in discussion forums and Sci-Fi media. I would argue that Von Neumann's ensemble interpretation does not differ far from CI other than that there is a transition from classes of systems to sets of systems in the semantics.
Ask Roger Penrose, or ask Sean Carroll, or ask Steven Weinberg about what's really going on in QM, and you'll get three different answers.
That is exactly the type of "which twin is really older" question that begins with premises contrary to CI. Ask rather "what is actually happening" where "actually" means in the sense of actions and interactions between observer system and environment. You get the same answers when you ask what happens in the lab.
Note that all explanations/definitions/deductions start with unexplained/undefined/non-deduced primaries.
This debate over interpretations is a debate over the choice of primaries. The CI stops at the operational laboratory actions and observations. These are the primaries and all else is explained in terms of them.
Classical physics stops at the objective states and uses these to explain "what's really going on" during the laboratory actions but the states themselves are left unexplained (well we actually go in a circle the way dictionaries do with definitions, and explain states i.t.o. laboratory actions as well... until you get to Everettes worlds, Bhom's pilot waves, and the per-Einstein aether currents). This circularity is fine in classical physics because all acts of observations are commuting Maxwell demons. This is an implicit assumption in assuming objective states as one's primary and all actions are evolution flows on the manifold of possible states.
The SR analog is the pre-relativistic assumption that you can always fibrate space-time into a unique bundle of simultaneous spatial snapshots (fibers) over the base sequence of time. Once time is relativized you must shift to primaries of space-time event points. It's still objective/classical since the primaries, while changed, have not changed type.
In QM under CI the primaries are changing type. The chicken and egg definition cycle of objects-observables, is stopped at observables because one notes there are more of them and they are more generally applicable than objects.
In the logic the set-inclusion lattice of subsets of states transitions to the quantum logic lattice of subspaces in Hilbert space. If you stick to only subspaces which are spans of a given orthogonal basis, you recover a classical logic lattice as a sub-lattice. You can embed classical descriptions in quantum. The thing is though, ALL the other subspaces have operational meaning. There are observables for these "states". There is more happening, more actions available, in a quantum logic lattice than can be expressed as a power set of a maximal set of primary states i.e. than can be expressed as point transitions between objective states. Quantum logic is a language of actions and it is a richer language than classical logic. This is why it is the natural place to start, the justification for actions are primary.
