As far as I know there is no theory that time does not exist. Yes of course there is a thesis that it does not exist but then there is a huge difference between proven theory and a thesis. Contrary to the popular idea, Einstein’s Theory of Relativity only proved that time is relative depending on the frame of reference of the observer. It was not a refutation of time but a deeper understanding of time and consequently the realization that time instead of being an independent factor, which inexorably goes from past to future at a set unchanging rate, was closely associated with space and the development of far reaching ideas about space-time continuum. Never, except in tabloids, was time as a measuring unit of our progress through life denied its role.
Newtonian physics for two hundred odd years kept time independent of all measurements. It was considered sacrosanct, unidirectional and my time was your time irrespective of how we moved in relation to each other. Newtonian physics did not take motion and its effects on time, the time dilation factor, into account. Nor was there a need to because science was in its infancy and human knowledge was coping with the explosion of scientific knowledge very close at home, on our planet only. That even with height, as on Mount Everest compared to the sea-level, time dilates a few micro-nano seconds was not something which could have been detected until well into the twentieth century. But what we measure as time, still dilates only with reference to the observer remains an incontrovertible fact. Its measurement by more and more sophisticated devices are still of paramount importance in every form of human activity.
The only problem and this has been pounced upon by sensational tabloids is the difficulty of defining “now” for a pair of observers, one stationary on Earth and the other traveling away from Earth at speeds close to that of light. Then each observer sees the other’s clock running slower than his or hers. But still time and its measurement are of prime importance for both. Time simply does not vanish except perhaps in regions which are classed as black holes in space.