I just tried publishing my newest game and it showed up in searches fine except I noticed the file isn't available in the download page or the Itch desktop app. Is this due to some review process or is there another issue I'm unaware of?

I have since gone back to draft status until this gets resolved.

It's hard to say without looking at your game page, but this could be from setting a release date in the future, or one of the visibility options at the bottom.