There’s a reaaaaaally thin line between (sincere) “We’re so worried about you, can we come in and chat?” and (menacing, oppressive) “We’re so worried about you, can we come in and chat?”.

And there’s a really thin line between having the choice to answer your door, and being compelled to for no legitimate reason.

Less menacing than either would be "can you come out to chat", or better yet, "come to the door to chat".

I'd let concerned EMTs into my house, but I'd rather keep concerned cops out.

If they say "can we come in?" it likely means they're legally not allowed to come in unless you invite them in. Which means they won't kick down your door either if you just don't answer.

But they can 'see something suspicious' inside, and then suddenly be duty bound to investigate. Do not open your door for cops, and if you feel you should, step outside swiftly and lock the door behind you. They will always find something if they want to, even it it was not there before they entered.

Nope, I'm not obliged to.

Sorry that's wrong. If someone knocks on my door unexpectedly I don't even bother to look outside to see who it is - I'm not answering.