Time is God's way of keeping everything from happening at once
...........sometimes attributed to Einstein or John Wheeler.
The times they are a-changing
Sean Carroll takes up the question of timelessness in physics, quoting Fotini Markopolou:
There are two kinds of people in quantum gravity. Those who think that timelessness is the most beautiful and deepest insight in general relativity, if not modern science, and those who simply cannot comprehend what timelessness can mean and see evidence for time in everything in nature. What sets this split of opinions apart form any other disagreement in science is that almost no one ever changes their mind…
Which unchanging character is certainly plausible if one doesn't believe in time anyway. Neither Sean nor Fotini has much sympathy for the anti-time warriors (whose original ranks included Einstein), though she would like to do away with geometry, gravity, and space instead.
As Sean points out, however, those in the timeless crowd still seem to be able to meet deadlines and read plane schedules. Time might be dispensable in physics but it's surely not in everyday life. One of our most fundamental applications of time is our sense that decisions still wait to be made about the future and still affect its nature. After all, if time doesn't exist, why bother to get out of the way of a speeding car? Or why bother to plan a physics experiment, if its results are already and always determined.
Our sense of time is thus a fundamental survival skill as well as completely necessary for doing physics or any other science. My feeling is that if time fails to appear in what you consider the fundamental equations of physics, so much the worse for the equations. The first duty of a theory of physics is to explain our observations of the world. We wouldn't tolerate physics equations which decided that gravity was unnecessary (sorry Fotini!) and we shouldn't tolerate equations that don't include time, not, at least, until one has an alternate theory which explains the same observations. The fact, for example, that general relativity is essentially timeless is a flaw - proof, probably, that the theory isn't quite right.
The fundamental question is whether there is any physical significance to a statement like "nature is timeless" or "time is an illusion." How would one go about planning an experiment that tested such a statement? Which invites the question of what "plan" (or any almost any other verb) means in a timeless world.
Whether we believe in time or not, however, we will continue to plan, decide, and otherwise act as though it did. And those who fail to do so are likely to quickly come to the end of their own worldlines.