You may well be right. However, I think the fact that two objects cannot occupy the same space at the same time, is more to do with the nature of objects, rather than the nature of time or space. The fact that 'an object cannot occupy another object' is surely the underlying and more fundamental truth, and that remains true even if we do not consider time or space.
And indeed, the movement of matter relative to space is not instantaneous, but does that require Time? Or, would it be enough to accept that the displacement of matter and energy in space takes place gradually and variably. ‘Motion’ exists, and objects have previous and predicted positions. The Big Bang started things moving with respect to the origin, but did it also create Time? Or did we invent the idea of time to help measure and compare motion, introducing 'past' and 'future' for practical purposes. Is Time simply an abstract concept that we invented to explain what we observe? We can move around in space, and we can touch and feel matter and energy. But Time is very abstract – we exist permanently in the present. We talk about the ‘past’ and the ‘future’ but we cannot access them from the present, other than possible by moving objects and energy of the universe back to previous or anticipated positions and states.
To make a loose analogy: we know that people have features on their faces, and we invented the notion of ‘Beauty’ for practical purposes, and so that we could, agree upon, compare, and talk about different arrangements of facial features that please us. We know that noses, eyes, etc. exist, but ‘Beauty’ does not exist in the same way. It is abstract. Could not Time be similar? We know that matter, energy, and space exist, because we experience them directly. But Time seems to be an abstract mental concept which we use to help us explain and compare the way in which matter and energy are evolving across space.