But a mezzanine is sort of silly considering no one lines up for an hour for a train like they do on a disneyland ride. All the stations in LA have these massive mezzanines too and they are just dead space. They are absolutely empty during rush hour aside from people charging through them to get to the actual platform, because it might take you several minutes to go from above ground, down the escalators, through the mezzanine, down a second set of escalators, and to the platform, at which point the train might be gone and you will sit there twiddling your thumbs for another 10 minutes, clogging up the platform for the next train since you should have made that first one. It's a huge problem at stations like 7th street in downtown LA, where the light rail platform is below a mezzanine but above the heavy rail platform, which bottlenecks transfers due to a lack of thought put into how many stairs and escalators should be made. Probably 1/5 days a week commuting in the before times would I actually make that transfer, the other times you are jammed up by people bottlenecked by the limited stairs and escalators and you watch the train go away without you.
Take out the elaborate mezzanine and make it a simple single level multitrack platform directly under the surface, and you might make the first train and be out of the station in less time, lowering the needed capacity of the station structure in the first place.
London Bridge Station has recently been rebuilt to have a mezzanine style area. It's the 4th busiest in the UK with 61 million entries and exits.
St Pancras and Kings Cross were similarly rebuilt.
It is a bit complex to describe London Bridge to non-Londoners, but simply put, it was 15 or so platforms that were about 20ft wide and 12 carriages long, with nowhere to queue and trains every few minutes, on every platform (and it's a terminus, not a through-station).
A mezzanine in a central city for passengers to wait is an absolute necessity. Doubly so if the station is a terminus, which all of the major stations in London are (I'm not counting Blackfriars or St Pancras).
For the non-English readers, there is only one North-South through railway line in London (Thameslink), and only one East-West (-ish) (Crossrail) and that hasn't opened. All mainline stations in London terminate, and then you have to use the tube to connect to another.
That’s a bit different though, as at London Bridge one is generally waiting for timetabled rail services whereas the New York City subway is turn up and go. A mezzanine is rather pointless if the same service is showing up every 2-5 minutes.
London Bridge tube station had to be rebuilt as well (not too add a mezannine per se but just to have enough space to hold the passengers waiting for trains), in the late 90s to house Jubilee Line but also to expand the Northern Line platforms. Even this 20 years later looks too small. Both these lines operate every 2 minutes (Jubilee slightly more than every 2 mins in peak).
To add to what the other commenter said - Bank station (the busiest on the London tube network) is getting rebuilt too, to add a "mezzanine" type area for the Northern line. Through-trains, all to the same destinations, are every 2 minutes and the platform is frequently closed due to overcrowding. To the point, in fact, where there's often a one-way system enforced in the station, or people are kept at street level.
Ultimately it's usage levels and capacity. UK stations don't have the capacity for the passengers, and often the capacity on the trains either.
I know this station and don't believe the mezzanine is the problem. There are two trains running perpendicularly. The "mezzanine" is the first floor with the blue line platform. Yes, additional stairs/elevators could help, as well as timing changes.
It's simply one of the busiest stations due to its location.
The huge mezzanine in the other stations such as at Pershing is mostly wasted space however.
In LA, you're going to be happy about those mezzanines once Measure R construction fills in your network. In infrastructure projects, what you see today is a small part of the plan.
Take out the elaborate mezzanine and make it a simple single level multitrack platform directly under the surface, and you might make the first train and be out of the station in less time, lowering the needed capacity of the station structure in the first place.