Evidence that Time is a Natural Phenomenon?

Question

Time is a tool we've invented to measure changes of state. What evidence do we have that it is a natural phenomenon?

Answer 1

Things actually do change, and change happens naturally.

The idea of time is inseparable from the idea of change. When we say that we are measuring how much time some process takes, what we actually mean is that we are comparing the rate of the process to the rate of some "standard" process (e.g., the rate of the Earth's daily rotation, the rate of vibrations of a quartz crystal in a stopwatch, the rate of oscillations of microwave energy emitted by cesium atoms under carefully controlled conditions in an atomic clock.)

Answer 2

A clock is certainly a natural phenomenon. A good clock measures time in at least two ways: a metric (duration) and a phase (when). But what is a clock? What does it actually do? I would suggest everything is a clock if we get loose enough with the defintion, some are just better at their job than others. How many heartbeats passed while you wrote your question? How many degrees did the sun process in the sky? How many atoms of potassium decayed in the ambient air in you room? Your computer's clock is subject to the same laws of physics as everything else in your room and above, however it regularly passes through much more arbitrarily close to the same states than "non-clocks". That's the only difference. And via GR, all clocks dilate the same under gravity/acceleration, no matter what they are made of. If the notion of seconds is obfuscating the similarness of clocks and "non-clocks", a second is just "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom".

So your heartbeats are natural and are clocks, and clocks tell time. Time is a natural phenomenon. Time is change.

The experience of time passing and/or the flow of time is something else entirely, but it seems like you were focused on the time is change idea of time.