> […] picking your preferred language is an unskippable step of setting up a new phone.
This does not address, or more precisely it even prevents the notion of:
> I might prefer some sites in Danish, but others I'd like to have en English.
, i.e. what you set in your OS or UA (browser) is always one preferred and optionally some fallback languages. I second that granular per-site control would be preferable for me either: some texts I prefer in my mother tongue (news, non-tech encyclopedic articles), the rest, namely technical documentation I prefer in English. (Also, I have very bad experiences with localised GUIs, where 1. familiarity with English "original" vastly helps when searching and 2. translations tend to range from bad to hilariously comical.)
But I must admit that most websites serving of content in multiple languages, support client-side pref pretty well, so the situation is not so terrible, in my experience.
Yet, the problematic of content negotiation and user agent author (provider) information passing is interesting and I'd say even under-engineered, and currently terribly suffer from security (anti-tracking) constraints.
(Lately, there was some development in "Client Hints": https://developer.mozilla.org/en-US/docs/Web/HTTP/Client_hin... )