There's a big push on Swish which is a mobile banking app and only available to people with BankID - there are smaller places that are Swish-only which means you cannot pay with a card or cash but a Swedish-specific app. I am now with Handelsbanken so this issue is solved but when I first visited it was an issue!

But regardless of those more niche issues, did you try joining a Gym before getting a Swedish card/personnummer?

Nordea -> issues Visa Electron. That is not the same as regular Visa!

Swedbank -> issues Maestro, https://www.swedbank.se/privat/kort/bankkort/bankkort-maestr... (yes, the English page lists only a quarter of the available ones)

Handelsbanken -> issues Maestro, scroll down a bit

I was there a few year ago with both Mastercard and Visa in my pocket and the success rate was 60-70% for that combination (i.e. some one third of the places would not accept either). But steadily improving.