If Kaecilius and the zealots were able to "break out" of a Time Stone backward motion, why couldn't Dormammu as well?

Dormammu couldn't break out of the time loop because he does not understand the concept of time. Time simply does not exist in the Dark Dimension, which is why Dormammu is so confused when things start repeating themselves; he can barely comprehend what Strange is doing, let alone begin to fathom how to break free of it.

Unfortunately, there doesn't seem to be a clean copy of the script anywhere for me to grab the relevant quotes, but it definitely is mentioned in the film that time does not normally flow inside the Dark Dimension.

Another thing to consider is that Dormammu himself may not actually trapped in the time loop. The entire universe around him is. He is noted as a creature that lives "beyond time", and that seems to be taken literally.

During the time loop scene, he is clearly experiencing all the time loops and has full memory of each one of them. He becomes increasingly agitated with each loop. He does everything in his power to stop it (which seems to be just kill Dr. Strange in a lot of different ways). When he finally gives up, he makes the deal.

Likewise, the zealots are human, so they live in time and they were initially trapped when Dr. Strange reverses time. It's noted during the movie that magic is energy drawn from other universes, so it may be that the zealots pulled energy from Dormammu's universe which let them break from time.

The Ancient-One, when she showed Strange that there begins a world of Mystery lying above his senses, she explains that there is a lot more unexplored out of the world Strange thinks of.(Strange looking only through a keyhole).

Meanwhile, exploring the galaxies and stuff, she takes him to remote areas, the unexplored worlds (like one where mind and matter meet) and finally gives him a glimpse of the Dark-dimension, a world filled with hunger, where powers OLDER THAN TIME lye.

Coming back to the question, The Ancient-One gave us(fans) a hint of what Dormammu actually is, an entity for whom time is nothing, this situation also came up with Ego the planet,simply because years, centuries millenniums and even eons (Asgardian term) are literally nothing to what they have seen.

And moreover, Dormammu's realm is unaware of the concept of time. Strange uses the time loop to simply repeat a particular timestamp to tire off Dormammu and eventually give up, there was no possibility he could overpower him.