^^ Yes, all those things will happen whether we are there to see them or not, but that correlation and predictability you discuss are not present unless sentient beings are observing the activity. And yes, this is indeed "time as we know it," but if we weren't here it would no longer be as we know it, I think:
There is no force called "time" that causes that atom to release energy at a steady rate; to imagine an atom working from a strict a time schedule seems a bit silly on its face, I think. Instead, the atom is simply changing its state and releasing energy in the process; that that process appears to us to be occurring on a precise schedule is a result of us attaching the yardstick of time to it, and not one of the atom following a precise time-ordained schedule. By the same token, the earth's spin and the moon's orbit are not regulated by time, but by their relative movement and inherent energy involved (i.e., inertia and gravity); nothing they are doing is influenced by a thing or force called time. Time is not behind these things at all, but it is necessary for us to use time in order to make sense of the changes, and perhaps help organize our world by discovering a "precision" in the changes.
These things only appear to be happening at precise, predictable rates because we are here to say "Hey, these things are happening at a precise and predictable rates." If we were not here to say that, they would still happen anyway, with things like rates, precision, and time having no meaning (mostly because there would be no meaning to have, what with "meaning" being another sentient invention). And yes, because the energies released by decaying atoms and the movement of celestial bodies tend to be extremely uniform phenomena, we can actually use them as a foundation for calibrating the tool that is time.
As an interesting aside, I read a book recently called How the Shaman Stole the Moon, which describes how ancient peoples originally noticed that the sun and moon moved in precise manners, and from that they learned to make predictions about moon phases, seasonal changes, and even eclipses, all of which both helped their tribes (i.e., knowing better when to plant crops) and empowered the shaman predicting the changes because it seemed downright magical. It wasn't magical at all, though; just a primitive application of time as tool.
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