Atatürk reformed and secularized Turkey. He changed many things to differentiate the new Turkish republic from the fallen Ottoman empire. He, for example, changed the alphabet, abolished the Caliphate, etc. Why did he stick with the obviously non-secular Ottoman flag, then?

I'm not entirely sure why the flag is "obviously non-secular." The star and crescent's association with Islam is because of the Ottomans, not the other way around, and is a Turkic symbol. See for example these previous threads on the matter:

https://www.reddit.com/r/AskHistorians/comments/24d72l/why_is_the_star_and_crescent_considered_the/

https://www.reddit.com/r/AskHistorians/comments/5rm5ap/why_did_the_crescent_and_star_become_symbol_of/

If you had some other idea as to why it's obviously non-secular perhaps you could elaborate?