Proper and coordinate time referenced by “t” is not time itself. It is like drawing an imaginary triangle on a piece of land in your mind, and saying the earth is in the shape of triangle. t(rigangle)= the earth
Time must be defined under the empirical basis - i.e. what we can observe and see in the real world.
You could claim time is physical existence in physics terms, but you must understand physics is the most idealised subject in history. When even they are saying something is physical, they actually mean is, what they imagine in their head sitting in the classroom or labs, not the objects they actually see, touch, hear and feel in the world of daily life.