Moon 'sighting' is much earlier than the Islamic tradition. It's discussed in the Mishna, so it was at least debated in the 2nd century CE: https://www.sefaria.org/Mishnah_Rosh_Hashanah.2?lang=bi

The charitable case is that while something might be predictable, it's our conscious interaction that's important to these religions, not the theoretical knowledge.

The uncharitable case is that religions derive a lot of power from being able to fix time, and this demonstrates their sovereignty over people's lives in this sphere.