As discussed, the airlines are required to check your COVID documentation at check-in - based on the requirements as to where you're headed. As your ultimate destination was Amsterdam, it did fall on SAS in Newark. Yet, in our travels, many countries also require follow-up once you land. But not at AMS, where we've often simply walked to our parked car or rode the train home.
For example, at Milan Linate (LIN), we were greeted by Airport officials holding thermometers, and everyone's temperature was taken. In Porto and Copenhagen, we had to show our passports and proof of vaccination (Our QR Code) before we could enter the terminal area. Similarly, in Spain, before we could enter Seville's terminal, we had to present our completed SpTH form. And in the Netherlands, a completed NL Declaration Form must be presented by all incoming passengers. And of course, there are different rules when transiting - like we did in London last week.
Countries sometimes do require "extra steps" for entering passengers. This can extend to trains also, as in Paris when arriving on Thalys at Gare Du Nord from Brussels, we had to show our passports and QR Code (vaccine status) to police officers based on the platform. Although I have dutifully completed my NL Declaration Form each time I've flown, now I don't even pull it out when I land at AMS... as I've never had to show it. So it does seem that post-arrival can be rather random and impacted by current local events. As COVID numbers change, so - it seems - do the "entry " requirements at some locations. Unfortunately, for those of us that live in the NL, AMS has never been diligent.
And yes, the airlines aren't thrilled about being the "enforcers" of all the COVID rules during pre-boarding. Like your SAS experience, I've flown many airlines from AMS that have online check-in and unmanned "drop-bag" check-in options... and no one has verified my documents. At the same time, we've also used VeriFly (on BA and AA), Ready-to-Fly on KLM, and FlyReady on Delta, and a few other EU-airlines pre-registration programs to enter our covid documentation. Perhaps that will be the future for all air travel?