I think Swedish banks issue MasterCard or Visa as their standard debit cards, so it seems to me that it should work (although I didn’t have reason to try it when I was there). See e.g.:
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?
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.
But regardless of those more niche issues, did you try joining a Gym before getting a Swedish card/personnummer?