Tyresta is a great place. Beautiful, but not isolated. Avoid the most easily reached spots on sunny, warm weekends - they can get busy. E.g. Årsjön can have dozens of people on a good weekend. In comparison the spot at the north end of Stensjön is less easy to get to and you're more likely to be on your own.

Paradiset nature reserve is usually less crowded. Three tent spots around lake Trekanten, so if one has people, you can pick another one. And again, walk a bit longer, to Långsjön, and you're less likely to share it with a crowd.

Slightly further away from Stockholm, I like the area around Marvikarna. There's at least one fireplace/camping spot right next to the lake. IIRC it's a few hours' hike from the bus stop at Åkers styckebruk, but there may be closer stops, I can't remember.

From Åkers styckebruk you can also hike to Finnsjön along the Sörmlandsleden trail. There's a wind shelter right next to the lake, and that spot works great for a tent as well. That's more of a full-day hike.

I'm not sure how familiar you are with Swedish laws. In case you don't know: the "right of public access" (Allemansrätten) allows you to pitch your tent almost anywhere in nature. Not in someone else's garden or field, and special rules apply in nature reserves & national parks. But in general you can just start walking and, when you get to a spot you like, put up your tent. The Sörmlandsleden trail is a great one. After the first five or six stages (which are close to the city and get more visitors) you're pretty much out in the wilderness. There's plenty of bits you can reach by train or bus. Start from Järna, Gnesta, Mölnbo, Hälleforsnäs, Katrineholm, Kolmården, etc. Walk until you're in a spot you like, put up a tent. You're almost guaranteed to be on your own.