Second biggest city in the US is Los Angeles. Despite the entertainment industry, it is, like Birmingham, a major industrial center, and a single massive paved concrete megalithic structure. If you fly into LA, you're slowly descending over solid buildings and streets packed to the horizon for about 30 minutes on final approach.

However LA has a real tourism industry that you would deliberately choose as part of a vacation package. For an all-work no-play city (in the sense that it isn't as big a tourism draw as other American cities) that ranks in the second tier of size for American population numbers, it is probably Chicago. However Chicago is double the size of Birmingham.

To match population size with Birmingham UK (1.1 M), you have to skip down the US list past Houston, Phoenix, Philadelphia, San Antonio, San Diego, and you stop around either Dallas (1.3 M) or Jacksonville (985K), depending on whether you want the higher or lower bracketing case. All of these are kind of non-flashy cities that don't have the tourism reputations of the bigger cities New York (8 M) and Los Angeles (3 M), or smaller cities with larger tourism orientation like Las Vegas (660K), Miami (456K), New Orleans (364K), or Honolulu (342K).