You would need memory as well. Without memory you wouldn't be able to detect change over time, and hence not the existence of time. And without that you can't have awareness (as I understand the word).
The past light cone of an object constrains what past events can affect an object regardless of whether the object has awareness of not. So you don’t need memory to experience time, even if you need memory to understand it.
Measurement, awareness, and memory, are not required for time to exist. These are all things that make observation possible, but time could still exist even if the universe were just a fog of loose atoms bouncing around with no self awareness.