You are right that you will need to get a visa to stay in a specific country for 6 months. However, "Europe" has several different regions. You can be in the Schengen countries for 90 days out of 180. The Common Travel Area (UK, Ireland, Isle of Man, Jersey and Guernsey) is its own region. You can be in the UK for up to 6 months without a visa.
So what you can do is split your time between those two regions, plus the other countries in Europe which aren't part of those regions.
If you get a visa for a year's residence then you will also be paying local taxes, on top of the US taxes.