Thanks so much for the references. The paper Time from quantum entanglement... was very interesting, also quite technical, but I liked that inclusion of clocks. Any discussion of time that does not include clocks is not really a discussion about time.
The paper involves two time dimensions as rest and moving frame clocks and the corresponding Hamiltonians use optical rotation as a nice little clock. Since time is all about the two dimensions that clocks measure, it is ironic that the conclusion is that there is no time dimension for reality, only spatial dimensions.
I completely agree that it is the four-space of GR that keeps gravity and charge forces from unification, and it is very likely that spatial clocks can keep time without the extra dimension of time...but it is also true that clocks do not need space to tell time.
Clocks only need matter and action and so time can be two of the dimensions from which space emerges. Clocks necessarily have two dimensions; a period or tick rate and a decay of the period over time. The polarization clock ticks with the period of the light frequency, but that is somewhat lost in this paper. Moreover, the decay of the clock is set by the stability of the period over time, also a point that the paper does not cover, the Anderson deviation.
The paper does do a nice job with entanglement and the rest and moving clocks can therefore interfere with each other, but the key to unification is with the two dimensions of time and not with space at all. Note that you talk about time travel because you think of time like space. Since time is simply a way to know objects are separate from each other, time travel has no meaning without an a priori and implicit assumption of space.
A space that emerges from the action of objects in time means that there is no sense to the notion of a journey back in time and the decay of each clock's period points the direction of time. Time is primal, not space...