Correct, the possessive case does it. It's the same for other phrases, and actually works pretty much the same way in English, too:
- kattens svans = the cat's tail (without "the")
- svansen på katten = the tail of the cat (with "the")
And similarly, you could say Sweden's people, although that wouldn't be very idiomatic.
At least in the southern US, the English word "folk", while not exactly synonymous with the Swedish cognate, is certainly often used in the sense of people. It is not unusual to hear things such as "you folk from Alabama" in the same way one might say "you people from Alabama". And so "Swedish folk" ought to be a legitimate translation of "Sveriges folk", IMO.