I'm not aware of any city in the world where someone can purchase a 3000 Sq ft+ property with a two car garage for less than 4x the average annual family income and have access to amazing public transportation.
Everywhere has busses, trams, trains and cities have subway.
If you really want a huge home and want to 'ride public transit' - then it's possible to get one near or by a bus route, or a short walk from the station at the village.
Or you can buy a little further out, and drive 2 minutes to the village and catch the train when needed.
"Good transportation requires high density or or high property values to pay the taxes required to build it."
No, it just requires foresight, social buy in and a tax base.
Almost everywhere in Europe has really good transport - and it actually is possible to buy big homes, very few want them.
>Almost everywhere in Europe has really good transport
No, it doesn't. If your bar for good public transportation is a bus route and a drive to a nearby train station, then the US has good public transportation as well because every city has buses.
Good transportation is within the cities like Paris, Vienna, Rome, London. Trains run at least every 10 minutes and can get you nearly anywhere in the city within a half hour. It has to be good enough that it's actually faster and more convenient than driving. All of the houses on these train stops are very expensive or very small (or both if you're in London).
No garage, a carport is enough for the few times a year someone needs to park a car there.
Bus stops every 90 minutes, so not exactly amazing, but good enough for the occasional trip to the city.
Note that bus service doesn't need lots of tax money to build up infrastructure, so it can be built out into even sparsely populated areas.
If that's your bar for good public transportation, then the US has amazing public transportation.
Also - Germany is quite a bit smaller than the US so property prices are an issue.
There's no reason there couldn't be a 'Frankfurt, Maine' - with large homes a little further out - and still cheap.