The clipboard is the only way to get rich-text directly to Notes. Notes does not support direct integration via rich-text on either platform.
On Mac, you can script note context via HTML, like the “Markdown to Notes” example in the Notes integration guide, but it’s pretty wonky, honestly. You cannot get rich-text that matches the default rich-text in Notes app that way, so I wouldn’t really recommend it, and there’s not equivalent on iOS.
Not even Shortcuts can send rich-text to Notes on iOS.