It DOES happen with car mechanics. Leonard Green LP owns Caliber Collision, Carlyle Group owns Service King, and Icahn Enterprises owns AAMCO, Pep Boys, and Precision Tune Auto Care.
While I don't have a specific example in my head about restaurants and grocery stores, I have to think that there are steps in the value chain that are absolutely owned and influenced by PE. Logistics? Real Estate?
Real Estate is a great example: most commercial real estate is held inside PE portfolios via Real Estate Investment Trusts, and lease rents are an extremely non-trivial portion of business costs for EVERYBODY, espcially grocery stores and restaurants.
But I agree about the many social and regulatory complexities regarding health care specifically. Often that complexity is what makes industries like health care such a lucrative investment for PE as it can be easier to hide incentive structures.