As European, that has lived across multiple countries, that only applies to the lucky ones able to afford living close to the city center.
Also healthy enough to be able to walk stairs, as very few places care about people with disabilities, or carrying stuff that is a pain to transport across stairways.
People visit the touristic centre of the main cities and assume we all enjoy nice public transport systems.
> to the lucky ones able to afford living close to the city center.
Which is also, to some extent, the reality in the US as well. Some number of the "city centers" have better public transport and/or walk-ability [1] available than what is available just outside those city centers.
One big difference in the US is the massive land area difference as compared to Europe means there is a huge amount more land area (and therefore population) with little to no public transport or walk-ability available and a car becomes mandatory rather than optional in those areas.
[1] It's not perfect, I'm sure there are plenty of city centers in western states where even the city center itself is so spread out that walk-ability suffers and that a car tends to become more necessary.
UK transport is much worse than the continent. London is fairly well served but elsewhere not so much, especially not the countryside. The trains are very expensive (even with an old person's railcard) and the buses are often irregular or non-existent in large areas of the countryside.
There's definitely a lot of truth to that, Europe is not a monolith in terms of transport infrastructure.
On the other hand, it's hard to overstate just how radically car-centric the majority of the infrastructure in the USA is.