Background: I work for a PE-owned company and I have friends in PE (associates up to MDs).
On your second point: LBOs aren't the only tool in the toolkit, and it's not as popular as it was decades ago, so I would lean towards the parent simply conflating "buying an ownership stake in a business in some capacity using other people's money" with the strict definition. Regardless, yes PE firms need to figure out how to get 20%+ IRR throughout a short timeframe (usually a 5 year holding/funding cycle) -- however this is through any means necessary. Philosophically, it's about increasing efficiency of operations and growing the business. In practice, it's financial engineering because PE firms do not have the operational skills to make any value-added changes to firms besides driving costs down.
Saddling a business with debt is reductionist. I've seen absolutely nonsensical financial structures that make no sense for a layman, but in practice end up "using the business' finances to 'own' (beneficially) the business" (see: at the most vanilla, the strategy of seller financing in SMBs). No this is not technically "putting debt on the books" but it is in all practical respects a novation/loan transfer that can leave the purchased co financially responsible for servicing any debt that was used in its purchase.
On your third point: what I wrote above can be used as context. It's not risk free revenue, frankly it's very risky unless you're in an inflationary environment where your assets will grow regardless of your business operations solely because the overarching economy is growing and you're riding a tailwind. However, it again boils down to financial engineering. It's not as simple as assets - liabilities = equity. The calculations used to determine valuations are so ridiculously convoluted. The amount of work that goes into financially analyzing businesses and finding "loop holes" that can justify higher prices is the core business model. The debt factors into it, but there's ways to maneuver around it through various avenues.
For example:
* debt-to-equity conversions (reclassification of debt as equity)
* refinancing
* sale-leaseback (selling company's assets to a 3rd party and using that money to pay down the debt, then leasing the equipment back)
* creative interpretations of what is actually debt (e.g. reclassifying real debt as a working capital adjustment or a "debt-like")
* dividend recapitalization (a nasty trick of loading the company with debt, paying that out as a dividend to the holdco, then selling the company at lower enterprise value. They still extracted value for their LPs/investors, despite the exit being lower)
* separating the debt from the operating company into a different holding company that services the debt