After re-ordering and choosing a faster shipping method, I finally have those tiny buttons:
They are definitely better than those "reset" buttons I used on nGame, but still a bit on the hard side. Also very clicky and with reasonable travel. The larger surface of the button itself makes a lot of difference.
Soldering them is a bit hard — the contact pads are not large enough to hold the buttons down with surface tension, so the hot air gun just blows them away. So far the most efficient way is to just solder them by hand, with the soldering iron, going from left to right and adding the buttons as you go.
Unfortunately, the 20 displays that I ordered (2x10pcs order) turned out to be 2 displays when they arrived (the seller admitted to a packaging error and refunded the difference), so I only have one prototype to play with and one unit for sale. More displays are on the way, though, so expect it on Tindie near you soon-ish.