London has terminus stations - the platform the train arrives on is where the train stops, and then starts on its return journey. Trains to your destination are therefore limited by signalling and traffic, typically a 10-15 minute turn around time. On average, in London during rush hour, you have a train to your home station every 20 minutes.
Perhaps in the US all major stations are through stations, but in the UK, especially London, we just don't have that, and mezzanines are at the very minimum, a safety requirement, because the platform your train leaves from, has other passengers waiting for other trains before your one.
Platforms are not large, and trains are very, very busy, and frankly, if you've only ever caught a train in the US you are in for a huge surprise commuting in the UK.
If the US ever adopted trains like the rest of the world has, the best time for building mezzanines, and huge stations is "yesterday".
To expand capacity beyond that would require reengineering the tunnels themselves. They go as fast as they can turn around and vent air as it is. You'd have to bore parallel tunnels to increase capacity, which would just mean adding more platforms and still wouldn't require a mezzanine.
This in a city of around 2 million inhabitants for the whole metropolitan area, which is also served by busses, trams, local trains, ferries and so on.