Modern physics can be mysterious. But not that mysterious.
It always astonishes me how people (including even professional physicists at times) manage to confuse themselves so badly, they start questioning the existence of even the most obvious things around them. Does time exist? (Check your watch.) Does space exist? (Check your meter stick.) Is gravity a force? (You tell me, after you drop a brick on your foot.) Is motion an illusion? (Ask again after you run head first into a brick wall.) Does matter exist? (Again, I could recommend various convincing tricks with the aforementioned bricks.)
Physics, modern physics in particular, may be abstract and difficult to learn, contradicting everyday intuition. But physics is not about questioning reality. It is about explaining how reality does what we observe.
So yes, time exists. Again, check your watch. As to how it works, we certainly learned a lot in the past century or so, with the discovery of relativity theory in particular and the realization that time and space are inseparable aspects of the same fundamental reality, the spacetime in which we live. And who knows, a future, more fundamental theory might even teach us that spacetime itself is an emerging concept from some even more fundamental element of reality. But, I emphasize, these are explanations. Not questions about the reality of something that we can directly measure with a simple instrument, such as a watch.