I scanned this thread fast so apols if it has been answered.
'If you climb at vx and you reach 2000' in, let's say, 5nm (at eg 60 mph = 5 mins) while in the mean time you reach in 5nm at vy, 1300' (at eg 120mph= 2.5 mins and 2000' in 3.8 mins but 7.6 miles) why in the hell are going to climb at Vy to gain altitude in the shortest time'
Because in scenario 2 you are going faster, so you hit 2000' quicker ( less time) and you are 2.5 miles and 1.2 minutes ahead of your mate(but if the mountain is at 2000' at 5 miles the point is moot). I think the vx vy numbers you quoted aren't quite that divergent.
Another way of looking at it is if vx=60 and vy=65 and you had a 60 headwind you will clear any obstacle at 60 ( vertical take off) but at 65 you will go up faster ( vertical feet per minute) but will move forward towards any obstacle.
It took me ages to understand the difference. Now you have 17 different views of why it's different . Hope one works out for you.