Yes, but time and distance can easily be considered equivalent without having to use a Minkowski diagram. Consider the term "light year" which is a unit of distance, but which also tells you exactly how much time it would take for light to travel that distance. Or consider the thought experiment commonly used in relativity, the "light clock," which is a hypothetical clock made of two parallel mirrors with a photon endlessly bouncing back and fourth between them. All I need to know is the distance between the mirrors, and I can tell you how much time it takes for each "tick" of the clock, at least in the reference frame of the light clock itself. Both of the above concepts work fairly seamlessly with the concept of a three dimensional space, with x y and z as the three spacial axes. But a Minkowski diagram requires us to neglect one of those spacial axes (usually not a problem) and replace it with a ct axis which, as you say, is where that darn time thing shows up. In that respect, Minkowski diagrams are not really all that different than any other type of graph which happens to have a time axis. But because relativity has some weird concepts such as relativity-of-simultaneity, which is well outside our everyday experience, some folks look for deeper answers in the Minkowski diagram. Some interpret them as showing that all time, past present and future, is one static "thing" and we are moving through it, as through a fourth spacial dimension. I find that to be going a little deeper than is necessary, but maybe that is just me.