I think a lot is fixed, but for a fairly short term. So you have roll the debt over (at the new market rates) every 5 to 10 years.
Closely related, you can read about the PE Zombie funds that cover a lot of this debt.
You've got it reversed, the debt BDC's and PC firms write is usually floating rate based off SOFR. That' s why they do so well in rising rate environments.
Now they often borrow at fixed rates by issuing bonds or preferred shares that pay a fixed rate so falling rates has to opposite effect on them.