You have just described Combined Sewer Overflow. Yes, that's how London works too. Whether it's "not quite as bad as it sounds" is a matter of opinion and also should take into consideration just how often such "overflow" happens and what volumes are involved.

For example, how many cubic metres of rain water makes adding one human turd OK ? A million seems fine, doesn't it? Like, who cares about just one turd in so much rainwater. How about a thousand, that's a swimming pool (not Olympic, but decent sized) with a turd in it, is that OK? Start to feel a bit uncomfortable with how much shit there is in the water? What if it's a hundred? Are you sure San Francisco, or London, can promise you their CSOs have at least that ratio?

At least on the US side the EPA's CSO laws have "just recently" ratcheted past the point that even "grandfathered" cities are having to make major accommodations for CSO overflow. If you live in a US city with sewers older than 1900 or so and are paying huge sewage taxes, it's likely because your city just built or is in the process of building massive CSO basins, uncombining sewers, and/or other costly capital expense mitigations. It's kind of interesting if you dig into your local projects what they are doing and why and how. (In a semi-related one of those projects my city's water company recently excavated a mid-1800s water main valve that was still in apparently active use up until just recently and it was pretty neat the photos and description of it in the local newspaper. Built to last.)

(Disclaimer: I vaguely worked on software projects that were CSO mitigation adjacent. Which at least explains some the interest in the projects.)