Hello chaps. Just want to close this out.
1) You cannot enter Sweden unless you are already in the Schengen area/EU national coming in via the Schengen area, or under one of the exemptions they've listed on the website. This means that if you're a US citizen in the UK, you're not getting in unless you're already in the Schengen area or, maybe, a UK resident (I didn't get to clarify that point, although it helped in my case, as I'll explain later).
2) Getting into the Schengen area is difficult as a non-EU national/resident.
With these two points clarified, happy to detail my case.
1) Was turned away at Heathrow. The belief was that I was allowed to fly direct to Sweden, but I could not travel via Switzerland. You are permitted to transit through Switzerland as long as you don't "enter" Switzerland. However, to transit to Sweden you need to enter the Schengen area in Switzerland, and that means that you need to enter Switzerland, so transiting through Switzerland to Sweden as a non-Swiss, non-Swedish resident is a complete non-starter. It wasn't even a conversation. Wasn't happening. And the agent read the notes directly from Zurich border patrol, which specifically referenced that you are not allowed to transit as a non-Swedish/Swiss resident even if going to Sweden.
2) This also applies to Germany.
3) There was a belief that if I switched to the direct flight, I should be fine.
4) The direct flight was cancelled.
5) I was able to rebook via another airline/Schengen country. And I confirmed before rebooking with the border police in that country that I could transit. They said I could.
6) Was allowed to board the flight to that Schengen country. And then went through passport control in that country. Breezed through.
7) On arrival in Stockholm, had to go through passport control again. My impression is that there's passport control for everyone, regardless of origin.
8) Passport officer said that technically none of us is permitted to enter the country, but for the Schengen stamp our passports bore. That is to say, if we did not have the stamps, we would've been turned away.
9) Further to 8, if we had flown direct from LHR, we would have been rejected at the Swedish "border" and sent back.
Luckily we were let in by virtue of the "loophole," but passport officer emphasized that we were very lucky.
I post this to say that I wouldn't dare try entering Sweden as a non-EU/Sweden resident. Even if you fall under one of the exemptions, you'll need very demonstrable proof to fly by.