You're overthinking it. Sweden made renting out apartments unattractive without all this sophistication, they just regulate the prices and keep them low.
Price controls leading to shortages and rationing is basically the microeconomics version of the First Law of Thermodynamics. It has had the same result literally everywhere it has been tried.
The solution lies in the supply. Always. If you want more people to have more of something at an affordable price, you have to make more of that thing.
[0]: https://www.bbc.com/worklife/article/20160517-this-is-one-ci...
They also can’t seem to get a sufficient supply of washing machines and instead have a laundry bureaucracy.
Sounds like you're just ill-informed? There are shared washing machine rooms in a lot of apartment complexes, although it is getting more rare in newer apartment complexes. You can usually book a slot and then you do your laundry. There is absolutely no shortage of washing machines though.
They do (with obvious drawbacks of anything illegal). And the black market prices are awfully high.