"Their first ask is to come into your house" is based on a quote Vice took out of context (and then blew up into a nice big sub-headline to give it a bit more 'emotional weight').
The second part .. if the police is concerned about you they ask to talk to you where they meet you. For the type of person where this intervention happens this will probably be their home most of the time.
I always wonder where people get this idea that the police (and law enforcement in general) are all crooks breaking/bending the law to get you, but will have a problem to get a warrant if that's really what they want. Maybe too many crime shows?
Bitter experience. Never underestimate the dishonesty of a cop.
Or rather, the lack of incentive for a person in a position of power, to not use that power for their own interests and at their whims.
No, police aren't all out to get you. But you'd be dumb even in Iceland not to have some basic situational awareness and consider how that interaction started out. You didn't come to them on the street, they sought you out at your home.
So if the police knock on your door open up and talk to them there. If they just want to talk you you they'll be happy to do that there. If they seem more interested in going inside your house than having a chat with you at the door perhaps they're really there for something else.
Hedging against that by doing something that costs you maybe 5 seconds ("let's just chat here") is just common sense.
The police can wait outside in case they’re needed.
Police are decidedly not mental healthcare professionals.
The police gets to do that kind of rather unpleasant jobs. Not their fault.
But some are. Some will plant drugs on you.
The one knocking on your door now, is it a good cop or a bad cop?
But in this scenario the police are unprompted at your door and their first ask is to come into your house.
If the police just want to talk to you because they're concerned about you personally they'll just ask to talk to you in a place they wouldn't otherwise need a search warrant signed by a judge to enter.