This thread was supposed to be about a link from Waterloo to Euston - it seems to have been well and truly hijacked - although since I'm the one that started this sub-thread about Waterloo East, I guess I'm the most guilty party here
But since this subthread is interesting, I hope no one will mind if I continue...
Why would you want to do it? It seems to me there are several reasons.
Firstly, you would be removing two bottlenecks: The point where trains have to cross tracks to get to the right terminal platform at Waterloo, and the similar places on the line into Charing Cross.
Secondly, you enhance journey opportunities for many passengers to travel with fewer changes of train. Most obviously many SWT passengers heading for destinations along the Northern line City branch (other than bank) require one fewer change of trains, which of course in turn will ease some station overcrowding. You will also allow people to travel directly from many destinations in SE London to destinations in SW London.
Thirdly (I accept this is a much more minor point) the overwhelming majority of passengers who currently need to walk between Waterloo and Waterloo East to make their connections will have a much shorter walk between their connections. (And from observation, that's I'd guess that's well over 90% of the people who get off at Waterloo East). In short, it solves the problem of Waterloo East station being in the wrong location for almost everyone who uses the station.
And in terms of ongoing cost - you will have managed to closed two entire stations without that much loss of service - which I would have thought ought to be good on efficiency/ongoing cost grounds.
Set against that - you'd be forcing one additional change for South Eastern passengers whose final destination is Charing Cross - they would need to change to the tube at Waterloo. I'm not clear how many people that would affect.
Now in terms of travel opportunities opened up, that's certainly less than the opportunities created by - say - Crossrail or Thameslink, since it doesn't get quite into the heart of London. But then again, the amount of building work required would appear to be vastly less (no tunnelling). Whether the benefits are enough to be worth the cost is something that I don't know, though to my laymans eye, it looks plausible.