Sweden most definitely should be added to the list of countries with exemptions. I am an American who successfully entered Sweden to be with my long-term Swedish partner. The best chance of getting in has to do with the right documentation and the hardest part of the journey, surprisingly, was convincing the airline to let me board the plane.
I flew with Lufthansa out of MIA with a layover in FRA final destination in GOT. When I first arrived to check in, I was told I could not board. Luckily, I brought plenty of documents to plead my case. Here are the documents I had with me:
1) Negative PCR test taken within 48hours of scheduled departure (Germany's requirement for transit)
2) Copy of flight scheduled for 90days after arrival in Schengen Area
3) Germany's transit exemptions/requirements printed from their government website (Just in case the airline thought I could not transit through FRA)
4) (Important) IATA's written entry exemptions for Sweden printed from the IATA website. I then highlighted the section which states "family connections are exempt: to know more about family connections please visit polisen.se"
5) (Important) From the polisen.se website I printed and highlighted each section which referred to me being a family connection to my partner and therefore exempt from the entry ban. There are multiple sections which state that long term partners are allowed to enter the country. I printed out everything that referred to my case and highlighted / tabbed the most important sections.
6) Proof of Relationship- this was done with screenshots of pictures we had taken together since 2017. I included the location and time stamp for each (this can be done on iPhone photos app)
7) (Important) Letter of Invitation- This letter was written by my partner which stated that he is confirming our relationship and that we intend to enter into a long-term cohabitation relationship when I arrive in Sweden. It included his person number, phone number, residence address, as well as his signature.
8) Copy of Swedish Partners ID card and Swedish Passport
It took me nearly 20 minutes to convince the airline to let me travel. The documents that I had with me helped me get through. Take all the precautions necessary and bring any documentation you feel is important. If they still decide to deny you boarding, the polisen.se website clearly states that the airline can contact them to confirm exemptions to the entry ban.
YOU DO NOT NEED A SAMBO VISA TO PROVE INTENT OF GETTING MARRIED OR ENTERING INTO A COHABITATING RELATIONSHIP
When I was asked by German Border control if I had a visa to stay in the Sweden long-term, I told him "We are applying for the visa when I arrive. If we do not have the answer before 90 days, I have a copy of my flight out of the country on the 90th day". He said "OK" and allowed me in.
I then boarded my flight to Gothenburg, Sweden and arrived without needing to go through Swedish Passport Control.
I hope this information helps as many as possible!