Yes, this is the crucial distinction. (I wish that articles criticizing PE were framed in terms of LBOs + bankruptcy-law instead, because that's the root of the policy problem.) Corporations can go bankrupt without risk to the human beings who are owners/investors in the corporation.

Note that from the lender's perspective, the risk is the same and in a perfect-information universe could be mitigated by charging higher interest. The problem for society is the externality that the business's services get worse.