(This is going to be way too much opinion... but I'll post it anyways. Feel free to comment)
Who says there is a monopoly on progress?
There are countless examples where individuals came to two different conclusions, and both called them "progress." This happens all the time throughout history.
In the past decades, it has been safe to say that Western culture has a monopoly on progress, because any attempt to progress in a different direction was overshadowed by our progress. After enough attempted failures, other cultures "gave" us the word. If needed, culture will invent a new word to describe something which supersede our "progress."
As for modernity...
This is more of an opinion question, as the word has many meanings in different groups. From your description, relying on empirical evidence rather than scripture, I can offer my opinion: modernity is the result of trying to describe the world in indisputable terms.
All words are subject to linguistics. Modern linguistics suggests that words themselves rarely have a meaning to themselves -- they are defined within the fabric of culture. If the culture is lost or changed, the meanings to words become vague.
I like to reference the kashrut rules for grasshoppers, because it is a religious example which is not very conflagory. There are four types of kosher locust: Arbeh, Sa'lam, Chargol, and Chagav. These types were well understood in the time of the Torah's creation, but their meaning has been lost to time. Jews seeking to honor kashrut law are trapped by this loss. Many Rabbinical traditions will forbid eating of any grasshopper, just to be conservative. Others allow the eating of grasshoppers only if they are a traditional food. It isn't simple!
This issue is less apparent in non-writtten religions, because the religion is not bound to historical wording the same way a written religion is. A non-written version of Judaism might be free to replace Arbeh with Nomadacris septemfasciata (currently believed to be the most likely species match to Arbeh), but a written religion must hold to the oldest written word.
This is both the greatest strength and greatest weakness of a written religion
Science seeks to resolve this issue. It identifies the issue to be the connection between the culture and the language. If they could use a language which is universal to all cultures, science could never be misinterpreted. In order to make this work, they were forced to pay a grave price: All scientific language had to be built from repeatable empirical trials. There was no way they could base anything off of a cultural phenomena (such as "virgin birth" or "a flaming bush") which occurred once and nobody can predict if or when it will happen again.
Thus science is formed around repeatable experiments. Many of the fundamental ones, such as those needed to comprehend the meaning of "force" or "momentum" can be done on a playground in a few minutes. Others, such as chemical reactions, can be done in a lifetime. One can identify the element we call "sodium" by identifying potential sources of salt (NaCl), and then running them through a battery of tests which are designed to isolate "sodium" from other elements.
The purpose of science is to leave behind not only a model of our world, but a set of instructions for which a future civilization could resuscitate our scientific language and leverage the body of our work.
So where does modernity fit in?
Science does not define a complete culture, as best as it can tell. Because it is quite independent of the observer, it is extraordinary at explaining events that occur far away from the observer (such as in a test tube). However, there are a set of topics which are so dependent on the observer that science's own linguistic choices leave it trapped. Virtually all people will agree that "morals" is a meaningful word. Often they even agree upon a one-word translation between "morals" and their native language's word for it. The science of anthropology recognizes "morals" exist, but finds itself at a loss to try to build a model of how they work.
I would say your term "modernity" is a term to describe the culture that tries to make sense of these holes science cannot grasp while readily bending to its will if new science comes along and conflicts with modern theories. It is pressed upon other cultures because we fear that, if we do not, science will cut that culture to ribbons. I do not state whether this is good or bad, it is simply what we do.
The current definition of "modernity" is simply the best culture we have come up with so far with science at its back. There are well recognized holes still, but we haven't found a better solution yet. Consider the "modern" phrasing "the theory of evolution is true." That phrase is actually of indeterminate validity. A more accurate phrasing might be along the lines of "The current theory of evolution fits the data we have collected, and we have rejected the most promising alternate hypothesis with a R^2 value of .99999." (numbers are nominal). A scientist will claim "when you say the first phrase, you actually meant the second," and may even use the second phrase in academic literature, but the first phrase is much easier to say and is close enough to the truth to convey a useful idea. Unfortunately, that phrase has also invoked religious wrath, because the wording is the same wording religions use. I don't think wars have been fought over this... but it's getting close.
All in all, modernity could be seen as our effort to reign science in before it destroys us. As science pushes outwards, we provide the best cultural guidance we can to try to prepare other cultures for the onslaught. There is no reason to believe it is all correct -- there's clear places where "modernity" is doing things to us we do not like. But anyone who refuses modernity must deal with science in their own way.