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Regarding saddles, the basic rules note:

Saddles. A military saddle braces the rider, helping you keep your
seat on an active mount in battle. It gives you advantage on any check
you make to remain mounted. An exotic saddle is required for riding
any aquatic or flying mount.

As a comment to this question, one commenter wrote those statements to mean that saddles are required for flying mounts. Another commenter wrote that his interpretation is that saddles are not required for flying mounts - exotic saddles only confer advantage when used.

Are saddles required for flying mounts? Has this been clarified anywhere else in rules, adventures, or comments by the game designers?

The section of the rules that you quoted is pretty clear that exotic saddles are required for flying mounts RAW. I think the confusion is coming because that paragraph describes 2 distinct kinds of saddle (Military and Exotic) without highlighting when the focus shifts from one to another:

Saddles. A military saddle braces the rider, helping you keep your seat on an active mount in battle. It gives you advantage on any check you make to remain mounted. An exotic saddle is required for riding any aquatic or flying mount.

The first 2 sentences describe the Military Saddle which grants advantage on checks to remain mounted, while the 3rd sentence is talking about the Exotic Saddle which is a different item and is required for riding aquatic or flying mounts.

If you look further down the page, you'll find a Saddle Table that makes it clear that the "exotic saddle" it refers to is a distinct kind of item (which costs 60gp vs. 20gp for a Military saddle), not just a strange kind of saddle.

Based on this:

An exotic saddle is required for riding any aquatic or flying mount

I would say its pretty clear that an exotic saddle is required for riding an aquatic or flying mount.

As the DM, I suppose you could impose or require Dexterity-based skills/saves to "hang on" to a flying mount.