Good eye on the missing "recessed push-in thingy" bpp,. but it's not broken or missing, it's just not visible.
What you can't see in the picture --the "recessed push-in thingy"-- is the "stage 1" booster, the shorter lever arm that you use first (it's folded in toward the right). What's in the pic is only the longer "stage 2".
The corkscrew works fine.. but your mention of the spring did get me thinking, though.. spring, spring.. and FINALLY I figured it out.
I realized that the metal strips act as a very mild spring to push "stage 1" out from inside of "stage 2" as you unfold stage 2 and prepare to use the item. In bpp's picture the "recessed push-in thingy" is really the back of "stage 1", and that rounded rectangular hole is where you can manually push stage 1 out from stage 2. My version has that hole, as well, and that's also where the ends of the tweezer/spring lie when the strip is inserted all the way.
I don't know if you can see it, but there is a little bend, a tiny "lip", at the apex of the tweezer, that conforms exactly to the lowest edge of the stage 2 lever.. a little bit of a "stop", I guess, to keep it from sliding in too far. In my photos, I have it facing the wrong way; the apex should be on the left.
THANKS for helping me think this through and solve the mystery. It's quite a nice little touch for such a gizmo to have.. but it is kind of fiddly and I can see why they might have gotten rid of it, probably to save production costs.
The "butler's friend" is much larger (in fact, that's what cantlogin posted). I had never heard that name before, though.
Monopol Ah So Wine Opener for Sale
This Twist-Style double-pronged Wine Opener is also known as "The Butler's Friend", or, "The Dishonest Butler", because a butler, so-inclined, could remove the cork without drilling a hole in it, drink a little wine, and replace the cork, undetected!
These Double Pronged Cork Extractors are often recommended for bottles with dry corks, or older bottles with fragile corks.