You are suggesting that Time is an emergent property which I don't disagree with but I have to ask . . . emergent from where? These are fundamental properties of the universe. Emergent or no, where do they have their roots?
Discussion
i think that matter is emergent, and time is primal, i agree with your idea that time is actually the first dimension
in my visualisation of what our universe is, it is like an artifact that starts as a singularity, and grows (this is reducing the three dimensions down to two) as you go forward, the diameter of the space component continues to increase, at a rate that produces the speed of light, so a square
in this model, energy, aka photons/electrons/beta/alpha etc particles, have two components, or first dimension of time, second dimension of space, thus they form cyclic spinning patterns and the harmonics of their spin rate determines the way that they interact - such as the different ways that different speed photons interact with the different kinds of atoms, which are further complex configurations of primary particles that are caught in orbits with each other, which is also why they can be broken apart when you can push them close enough to the speed of light and into collision, which also can happen by gravity, such as inside a star
also, i didn't make something clear... energy, aka radiative particles, are not actually moving relative to the space matrix, they are like ships with sails, and are pushed along with the expansion force which they convert into forward motion because they are rotating in two dimensions, whereas matter particles have three components of rotational motion, which means they resist the expansion and appear to be staying still, according to themselves
it is the angular orientations of the energy particles that moves the matter particles into new configurations with each other
it is also like teh paradox of the red queen "you have to keep moving forward as fast as you can in order to stay in the same place" because the matrix (i think it was a chess board) is constantly expanding