Is time dependent or independent?

It is a mistake to think of time as an independent 'thing' that somehow exists and moves or doesn't move. Back in the early 1900s, Hermann Minkowski showed that there is no such thing as time or space, but rather a 4 dimensional construct called "spacetime". Like a two-sided coin, spacetime exhibits properties of both space and time, sometimes more of one than the other depending on the motion of the observer. A concept that Einstein used as the basis for relativity theory. But that 4 dimensional construct still exists whether observed or not.

We don't really know if spacetime began with the big bang, but if reality is truly made up of somewhere between 5 and 26 dimensions (as predicted by some string theories), then all of those dimensions existed before that initial expansion of our spacetime from Plank-sized to macro-size. And there was certainly no one around to observe it happening.

Time is independent.

Time started with the big bang about 13.8 billion years ago.

For most of that time there was probably no living thing in the universe.

In fact it took 13.8 billion years for intelligence to appear on this planet (planet Earth)

Life could have appeared before us on some other planet. We don't know when or if that has happened. But we assume life now exists somewhere other than on Earth.

And as you can see -- time kept right on going all that "time" before we turned up -- we are little "Johnny come lately"s.

And if a comet crashes onto earth tomorrow and all life on earth is killed, the universe will go on without us - until the end of time.

Time is a really difficult concept. Time is not clocks. It's really change. That is, time is (IMO) an emergent property of change (matter or energy). If there is no change, there is no time. So time is dependent on change.