This mostly correctly describes a leveraged buyout (LBO). LBOs are done by LBO shops, a type of private equity (PE) firm. Not VCs. (VCS do venture capital, a different type of PE.) And LBO debt isn’t “pushed” onto the company’s books, it’s never on the sponsor’s (LBO shop’s) books in the first place to any material extent.
Private credit, on the other hand, involves e.g. Blue Owl borrowing from a bank to lend to software businesses, usually without any taking control or equity. It’s fundamentally different from both LBOs and VC or any private equity inasmuch as it doesn’t have anything to do with the equity, just the debt. (Though some private credit firms will turn around and lend into a merger or LBO. And I’m sure some of them get equity kickers. But in that capacity they’re competing with banks. Not PE. Certainly not VC, though growth capital muddles the line between what is VC and other kinds of PE or even project financing.)
Doesn't the LBO shop still need to pay off the debt, technically speaking? AFAIU the company's assets (hospital in OP's example) are used as collateral in a credit agreement between the LBO shop (as the hospital's new shareholder) and the bank. But unless I'm mistaken, this is not exactly the same as the debt being on the hospital's books and the hospital having a credit agreement with the bank. (For an increase in debt on the liabilities side of the balance sheet there would have to be an equal increase of assets on the other side. The hospital didn't receive the cash, though, nor does the hospital suddenly own itself.)