Chrome and/or Firefox don't ask me to save my Home Assistant password on the login screen

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@Madivad

It even affects the app in ios, so it leads one to conclude it has been purposely included by design,

Nope, just an issue with browsers/password managers apparently.

Password managers will need to update their extensions to work with ShadowDOM.

It was covered as an issue in the Month of WTH:

WTH do password managers not work

This technically isn’t an issue with HASS, it’s with the password manager. See this comment from Balloob: https://github.com/home-assistant/frontend/issues/1622#issuecomment-418280967 https://github.com/bitwarden/browser/issues/713 just as an example. So probably won’t be fixed (at least not easily).