You've found the actual solution, it seems. I have done the same with other Alps switches, but it doesn't work 100% of the time.
The issue is some something that is easy to fix, as it is a problem with the way the tactile leaf is situated in the switch, and how much pressure is being applied to it. It may take a while, but it worked for me.
Be careful, though, to only put a single piece of paper in, and to have it fit as snugly as possible width-wise. Otherwise, you can bend the little wings on the tactile leaf.
Additionally, you can use some scotch tape as well (though I haven't myself). Apparently it works just as well.