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.
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.