PE is popular for one reason and one reason only: taxes. PE generally makes money on a trade called a leveraged buyout (LBO), where they take out a massive loan to buy a company. Because interest on debt is tax-deductible, going debt-heavy increases the take-home profits of the company (this is called a "tax shield"). Because the profits are higher, the value of the company is higher, and the PE firm makes money on their trade.
What this means in practice is that if you run your company sustainably (low debt, lots of assets). You become a target for a PE firm to attempt a hostile takeover of the company, all while claiming (defensibly, actually) to be doing whats in the best interest of the shareholders. So good companies will try to ward off these attacks by taking on lots of debt and going asset light to minimize the value gain a PE firm might have.
In short, both PE ownership and the brittle, debt-heavy nature of the American economy today can be traced to the tax advantaged nature of debt. For reasons I can't quite understand, nobody seems to be advocating for revoking this tax deduction. I can only surmise this is because everyone hates taxes.
Thank you for coming to my TED talk, your take home exam is a short essay on what you think the mortgage interest tax deduction (started in 1913) did to household debt.