What I don't understand is how the cost of banks repossessing these companies in case of default don't make the math unviable. Unless the company have a lot of fairly stable semi-liquid assets (like real estate) banks should be charging fairly high interest on these loans which would make most of these business unprofitable.

Which would increase the rate of defaults (if they are authorized in the first place) and in turn increase interest even further. I guess the PE is always maxxing out the leverage on every deal at _just_ the projected break-even point for loan repayment? But that leaves no room for error or changing market conditions which also increase the rate of defaults and so on.