The London sewer system existed before that time, and it was privately owned by several different companies but they were hampered by rules from local government.
And yes, competition in health care is Highly limited in the US. In every case, when a third party pays the bill (a single payer system/medicare or one with required coverage by private insurers) the cost skyrocket, while the prices for less regulated procedures go down. Consider lasik surgery, not normally covered by insurance but the cost has come down dramatically in just a few years. The American Medical Association also is a huge barrier to entry in the medical field and basically acts as an occupational cartel. Have you heard of a Certificate of Need? It's what you have to get signed off on by local existing medical providers, before you can open you own practice. Spoiler alert, they dont. This is some of what is stopping the market from serving consumers in the health care industry that desperately need more choice and less government.
And why are single-payer systems doing so much better outside the US?
https://mises.org/library/socialized-healthcare-vs-laws-economics