These businesses would simply stop existing if no purchaser came forward. Many first-world economies face a demographic cliff where boomers are retiring and there is nobody in the next generation who can afford to take over the local dentist office, the HVAC company, etc.
I don’t think that’s true though in all cases. Sometimes PE just offers more money passing the debt into the company they acquire. But smaller players would still take control.
You are forgetting that if PE weren't there to buy out local dentist offices at a premium, then these offices would sell at far lower price, making them affordable to the next generation.