Scientific enterprise is far older than the Greeks, who acquired their knowledge from the local Thracians.

Depending on how far back you're willing to go, it may as well have started half a million years ago.

Your question is like "How did Mathematics start?"

The first idea would be to say that it started with the need to count things. Yet if a smart animal can count to four, meaning that it's math system is comprised of "nothing, 1,2,3,4, a lot", is that truly mathematics? If it is, then math isn't just our thing and if it isn't, then where do you draw the line?

Same can be asked of science. Now to answer that basically it was the Milesian school is just part of the narrative of how Western science evolved as obviously there would be the "shoulders of giants" that the Milesians stood on too. Yet the idea that a) lets look at reality without religion and b) let's assume there are laws in nature are simply part of what I would call "Let's look at the World realistically".

And I'd say science evolved basically from a necessity, like mathematics. You simply cannot have people just by accident noticing stuff and applying it before the Milesian school.

It seems that our definitions are different. I agree with Popper that science is merely critical mythmaking. Atheism and naturalism are both ideas that weren't fully developed until the 19th century. I think it's a mistake to think that any of these two foundational ideas were incremental for the development of science. Definitely for materialism and other metaphysical philosophies that are held by many scientists, but not science itself. Many of the ancient scientists had God in mind when developing their ideas, although that didn't prevent them to do science.