Doubling the size of the lounge would probably be a fraction of the cost of building 15 and 16. As you have pointed out boarding would need to be controlled by staff as many people sinply refuse to wait upstairs and wouldn't change behaviour if the capacity and facilities are improved. Its not unreasonable to expect anyone of normal mobility to wait upstairs until the train before has departed to save £200m on building extra platforms. 12tph should be possible with better management and more suitable rolling stock. Major investment will be needed at some point but the current infrastructure is poorly managed. For instance it is staggering that there is not a "next train to Manchester Airport" electronic board.
It would be nice to think so, but I've been on 13/14 when trains have been so packed those standing near the doorways have had to get out to let others out, then get back in before boarding could start. Then half those on the platform can't get in and are left bewildered as to where to go next. If it was a late and overcrowded short formed East Midlands towards Sheffield it's probably a TPE over on another platform - where they'll probably find its 3 cars are already rammed full, so back to 13/14 for the next East Midlands. No wonder it's so often a milling throng of stressed humanity.
It amazes me that so many accept it's normal to stand for an hour on trains that are so often late, and they put up with that risk every day.
Longer trains won't necessarily solve the problems. Passengers dashing up and down to find coach A or F, especially those dragging wheeled cases to trip up the unwary, clash with those arriving from foreign parts unable to comprehend the mayhem into which they've been thrown.
Whichever way you look at it passenger numbers should continue to grow, so this isn't going to get better in a decade, far more likely it will get worse.
Of course reducing the number of trains would help, as would reducing the number of passengers. The challenge needs to be met by a range of measures, none of which is likely to be cheap or quick.