In layman's terms, why do we put the "time" in space-time?
To answer your point about why "spacetime" as opposed to space and then time - photons (and other massless particles) travel through spacetime on "null geodesics" - in other words they do not experience time at all. But they are not in some separate class as particles at approach ever closer to the speed of light travel on paths through spacetime that are ever closer to null geodesics.

In other words the four dimensional geometry is inextricably linked and "space" and "time" are a unified whole.

In theory humans moving also experience this effect but the speed at which we move is far too slow for this ever to manifest itself to our senses. But we do see its impact in high precision timing - such as with GPS satellites.