At the end of the day, they're just different ways of reaching the same goal: universal packages for Linux. I personally use them interchangeably depending on the application and use case.

There are some packages that definitely work better and are intended to be used and installed via your native package manager (if they rely on system libraries and whatnot). But using either a Flatpak or AppImage results in the same experience - in my experience. It's a personal preference.

I ran into the same issues, mentally, when trying out AppImages for the first time - but my attitude changed once I found and started using this tool: https://github.com/ivan-hc/AM

Usually the prompts are specifically for tracking cookies, not essential ones for login. Alternatives without cookies:

• URL sessions
• Tokens
• OAuth/OIDC third party
• Local/Session Storage (ditto - mind the risks)