You're right that the apparent symmetry is broken by acceleration(s!), and to show that I'd point to Michael Weiss's twin paradox equivalence principle analysis at
https://www.desy.de/user/projects/Physics/Relativity/SR/Twin... rather than rewriting it.
There is a subtlety not explicitly raised in the writeup, mainly that in General Relativity metrics do not superpose cleanly, in the sense of getting another solution to the Einstein Field Equations. We do not worry about this in the ultrasimplified twin-paradox model where the spacetime is flat in the sense that the Riemann tensor vanishes everywhere. However, if we want to consider the behaviour of gravitational waves with amplitudes outside of the weak https://en.wikipedia.org/wiki/Linearized_gravity limit, we are in a world of calculational pain.
Physicalizing this subtlety, if our travelling twin is travelling in our neighbourhood of the galaxy, it is probably in for a bumpy ride due to gravitational waves from nearby binary stars https://news.berkeley.edu/2021/02/22/binary-stars-are-all-ar... . We cannot easily extract how bumpy by adding in the uniform pseudogravitational field proposed by Weiss. On the other hand, we probably cannot quantify the effects of gravitational waves at all by simple adapatation of the other strictly Special Relativity analyses at the related Weiss link, https://www.desy.de/user/projects/Physics/Relativity/SR/Twin... (which lists among other the resolution in the minutephysics youtube link you provided above).
You're also right that the problem is one of reference frames. We are not obliged to use that of one twin as the spatial origin. In principle any will do, but some choices have advantages driven by features deliberately excluded from the Special Relativity twin paradox.
Let's consider the "(s!)" tacked on at the end of acceleration. We have not only that of the travelling twin's spacecraft engine, but also that which drives the expansion of the universe.
From within our galaxy we observe a highly spatially homogeneous and isotropic arrangement of extragalactic luminous matter (and cosmic radiation, locally) without distortions in the shapes of distant spiral galaxies that imply a spatially non-flat universe. The metric expansion of this, retaining bulk isotropy, gives us a preferred foliation (Wald's 1984 textbook develops this pp 92-93, but alternatively we could use Weyl's principle). Each twin is free to use a "cosmic fluid" observable (like the dipole-free temperature of the cosmic microwave background, which expands adiabatically), even while accelerating, to determine the https://en.wikipedia.org/wiki/Scale_factor_%28cosmology%29 . For example, each twin could consider the dipole pattern dT/T = v/c where T in the twin's proper time. Each twin can thus determine whether it is the relativistic traveller or not, even if it only wakes up occasionally and only long enough to look at a snapshot of the CMB. The travelling twin thus sees a clear breaking of the Copernican principle along the direction of its travel. Or more precisely, with respect to the bulk flow of matter and radiation in the universe, the non-travelling twin can conclude that it is effectively a Eulerian or comoving observer, while the relativistically-travelling twin cannot.
Moreover, the twins (and any third party) can use "cosmic fluid" observables to determine the scale factor when the twins are together at the start of the travel, and when they (or at least one and the other's remains) are together again at the end.
In this approach there is no paradox at all, there is only the consequences of one twin with a worldline with sections where the proper time is at a higher tilt to the cosmic time than the other twin's. We also avoid the difficulties in attaching a pseudogravitational field to a spacetime where there are gravitational waves of reasonably large amplitude, or relativistic stars and other massive compact objects.
We head into the land of apparent paradox by stripping out evidence of an expanding universe. We must also eliminate evidence of the aging of galaxy clusters through gravitational collapse (including the rate of star formation and the change in abundance of heavy elements). Indeed, we have to arrive in a setting in which neither twin can determine that it has departed from a point at which some reasonable generalization of the Copernican principle applies.
Indeed, the usual formulation of the apparent paradox gets rid of everything but the twins, so that one cannot even use Rindler/Unruh-like observables in flat spacetime, and this really emphasizes the "Special" in Special Relativity.
In that setting, as I said above, relying on the equivalence of being in uniform acceleration (even if it's instantaneous) and being immersed in a uniform (pseudo)gravitational field, is a reasonable way to eliminate the apparent paradox.