If the company had successfully paid back the loans then they would not have failed, but often when that happens private equity simply takes out more loans until the company fails. In such cases failure is both profitable and by design because they effectively sell the company to banks for more than it's worth while profiting from the difference.
This is incorrect.
"PE-backed firms are no more likely to default than other firms with similar leverage. Distressed PE-backed firms restructure more out of court, restructure faster, and are more likely to remain an independent going concern following the restructuring, compared to leveraged borrowers that are not PE-backed...Hence, PE investors do not exacerbate the likelihood of financial distress and seem to resolve distress more efficiently than other firms."
Furthermore, "companies that were previously owned by a PE fund that [have] exited within the last five years have nearly a 50% lower default probability than other firms in the sample, including non PE-backed firms with no history of PE ownership. Consistent with the findings in Harford and Kolasinski (forthcoming), this result suggests that PE firms leave their portfolio companies in relatively strong financial shape when they exit."
All in all, PE-backed companies default about 5% of the time versus 3% for population (Table 2).
http://portal.idc.ac.il/en/schools/economics/about/documents...
As to over leveraged, the most attractive targets for PE is when the companies assets are worth more than their stock. That's the opposite of being highly leveraged. Remember, profit and assets are two very different things and a company can lose money while still having a lot of capital.