Yes, you're right, the problem here in the UK is indeed overcrowding more than anything, but the problem is that it doesn't take Â£50bn to solve the problem and that Â£50bn is being spent such that it wont solve the problem.
For example, the line connecting the UK's 3rd and 4th biggest cities (Leeds and Sheffield) still isn't fully electrified, so you've got this absurd situation where it takes 50 minutes to travel the roughly 30 miles between them on 40 year old diesel trains that regularly break down, sometimes on the part of the line that is electrified and carries the East Coast mainline traffic to London delaying more major transit routes. This problem is repeated in other parts of the country like some of the lines around Manchester.
Then with HS2 itself you've got the further problem that it's not even clear what it achieves. You can save half an hour in practice from London to Sheffield for example, but the Sheffield stop will be at an out of town shopping centre from where it'll take you 30minutes (including walking/waiting time) to get a connecting tram, taxi, bus or train back into the centre meaning you lost any benefit of the increased speed. When the project was originally proposed there were two times listed - the theoretical maximum if the new trains ran full pelt from A to B with no stops and the actual times in practice with stops at each station. Nowadays all mention of the actual times have been deleted and only the theoretical times are mentioned by the government, but they're bullshit and will never ever be achieved in practice.
The issue is the busy lines aren't the East Coast line and the Sheffield - St Pancras Line - I've caught these many a time working between Leeds, Sheffield and London both off-peak and on peak. I've never once seen them full, and the only time people have ever had to stand for a few stops is when you've had a cancelled service and everyone's had to pack onto one train. Where we have actual persistent overcrowding is on the relatively short-haul intercity commuter lines.
So it's not clear what exactly HS2 is trying to do, it's a phenomenal amount of money to give you no real time benefit due to requiring new out of town stations from where you have to get back into town whilst failing to resolve any actual practical overcrowding problem and capacity isn't currently an issue on the existing route it will cover (any current capacity spikes, and future needs can be dealt with by simply eliminating or reducing cancelled services, and by putting more trains on the existing lines- there's still plenty of scope for that).
You could resolve the real local commuter overcrowding problems with only hundreds of millions - it's widely known what needs doing and relatively cheap and easy to do. Instead we're blowing Â£50bn on a boondoggle that solves no actual problems in practice.
There are some major rail projects that make sense- Birmingham to London does need major work to speed it up, but for the rest of the North, like Sheffield and Leeds it's mostly just electrification and maybe a few Sheffield line / East coast line trunk line to connect the two in case of idiots on the line or similar major delays (though I'm still convinced idiots on the line is better solved by sticking meat grinders or chainsaws on the front of the trains).
Finally as I say though, even if HS2 was the right choice, there's still the glaring question of quite how they're managing to make it cost many multiples more than things that are far bigger, more complex projects requiring far more expensive materials and far greater logistics. For the price of HS2 we can have two cutting edge nuclear power stations, two multi-acre aircraft carriers and still have change left to ensure every single household in the UK has fibre optic broadband. Whatever the merits of the project - if the MoD which is known for it's inefficiency can manage far more complex projects for a fraction of the price then something is very wrong with HS2.