(And note: on a policy level, this is fundamentally different from your neighborhood's bar having a conveniently broken ID reader: an online service can't meaningfully claim to lack the connectivity needed for ID lookups.)
Partial connectivity outages happen literally every day. Most people don’t notice. Websites like “down for everyone or just me”, or uptime monitors that monitor from dozens of different source locations, exist because of this.
The service itself can't say that it can't connect to you, because it's an online service and you're trivially connected to it.
You have a government SSO solution (2auth?) called NemID / MitID that is normally used for Banking[1], doctor or hospital appointment, library, student loans and what not.
But if you want to gamble on a Danish online casino you also have to verify your account with it. You can even set yourself on an exclusion list, and will be banned from all Danish online casinos. [2]
What is JSUT?
[1] No matter what bank you are at, same login, pretty neat