Is the idea here that saving the data from the cookies will speedup things, because requests become smaller? Or does it have some impact on caching or something else that I am not considering right now?
I would expect that unless you have exceptionally large cookies, the saved roundtrips from another TLS handshake matter more than the data transmission for the cookies.
It's a security matter more than a performance matter, although improved performance is a nice side effect.
For assets served from a third party (a CDN), you don't want to send cookies that might include secrets (a session cookie that could allow access to a user's account for example).
You can trust that a third party won't intentionally log or make use of any sensitive information in cookies but you can't guarantee it. Best not to send it at all.
I mean - if you separate your html from your assets security wise, that naturally means they need to be on different hosts, as you cannot really reroute requests before TLS decryption based on paths or any other indicator.
But the motivation to put stuff on a CDN would be to improve performance. If you put your HTML on your own HW and your assets on a CDN for performance reasons, you might want to check if that really pans out, because those extra roundtrips may kill all performance savings you get from the CDN.
