Still need to digest more of this, I agree with the general MUH idea, but the CUH, and the static bird description seem like... well, baggage.
Simply stating that spacetime is divided up into 3 spatial and 1 temporal dimensions overlooks the qualitative difference between what space and time are.
Space is a structure which defines relations within itself in a particular manner.
The use of dimension as a parameter, and the use of dimension in reference to some spatial parameter (this has a width dimension, that is a given distance away, etc) doesn't properly express what a temporal dimension is.
If a set of 3 spatial dimensions, analogous to the concept of directions, is described in terms of relationships and distances... then it defines the various manners in which those relationships can be adjusted.
The naive picture of 4-D spacetime is that all of the different states which our set of 3 spatial dimensions have contained/defined are contained in the 4th dimension, which many stick the label of time on and call it a day.
While concepts of extra spatial dimensions are possible to understand, yet can be quite difficult to picture. Further, the concept of systems with n observable dimensions are quite naturally handled mathematically... yet this still doesn't bring up the difference between a spatial and temporal dimension.
If you can say that something is located at a point in relation to the 3 spatial dimensions, it isn't as simple as adding another picture of the same dimensions in a slightly different state.
Time is a dimension, which is a degree of freedom, a way in which measurement can be defined in relation to other measurements of the same order.
It feels natural for me to describe it with the statement "Time is the direction Space moves through." As I realized that with all of my years studying curved spacetime and such in GR, I never actually realized that it isn't sensible to simply imagine space curving as literally as illustrations try to show.
You can't really say up was displaced to the left, can you?
So if up is warped or bent, then that curvature occurs within the direction of time.
This is something I deduced from SR and GR, but it is not stated in such a literal manner within the theories, nor often at all. Though considering the theories with this in mind, it seems a very natural description.
Where was I going with this ramble...
Oh yeah, the CUH, it does not seem proper to limit the overall mathematical structure to a specific state, as it seems to be ignoring that the sum of possible states that the structure could take is the 4 D spacetime as a whole.
The selection of a particular history is a single aspect of that, which would appear to imply that you would need at least one more dimension to represent the possible histories.
The value "one" applied to a spatial dimension, is not the same as the the value "one" applied to a single temporal dimension.
A spatial dimension describes the possible locations along itself, and a set of spatial dimensions enable relationships to be defined between those possible locations in terms of the other members, and the rules they obey.
A temporal dimension describes the possible states that a spatial dimension itself can assume.
The difference along a single direction is positive or negative from another point.
The different states of a direction curved across time can only be defined in relation to the other states, without a particular positive or negative sign.
Time needs to be handled better, I think most can agree, but the further one immerses themselves in the examination of these concepts, the harder it becomes to ignore ideas like the MUH.
Max Morriss