Another way of putting that: This isn’t a vector at all, it’s just a direction. Treating it as a vector gives rise to silly statements like “one second per second”, which is yet another way to explain that it’s magnitude 1… because it’s a direction.
I mean like yes you can measure time and space with the same units in the way you suggest but then the concept of velocity changes as well.
But also our own personal velocity is stationary. We (AIAU, IANAP) always perceive our own velocity vector as (0, 0, 0, 1). When we undergo acceleration it only ever affects the directional components of every other part of the universe, not our own experiential frame.
Moving your point of view from one inertial frame of reference to another is easy enough, but there should be some overarching mathematical construct that can model all the inertial frames and their relationships at once. Phenomenons such as energy, mass and acceleration should be easier to understand within it.