Private equity snaps up disability services, challenging regulators

Is it correct to understand private equity as transferring service quality to the initial set of customers (who were subsidised) from the new customers who have to give up quality to make the whole venture feasible?

No. Private equity means investors who buy equity in non-public companies. That's it. (It used to imply leverage, but that distinction has blurred away.)

In popular discourse it's close to a meaningless term.

You are technically right but I wanted to understand the bigger reason of why this happens:

A company runs well. But then they sell to a private equity. The quality goes down.

This is the common critique against private equity.

There are a bunch of different patterns (not all of which reduce quality), but usually when a company is sold to PE and quality goes down it's because the company was in trouble, either because they weren't profitable (or were, but only on paper), or were in a liquidity crunch, or were facing down imminent changes that would make them unprofitable.

People are often comparing to a situation where the company continued doing things that weren't sustainable long-term.