GEEZ that's enough reason to give the box back and get a handmade one. And yes, i notice some reeds don't play. Those "leathers" are the one-way valve so air only goes in one direction so only that note plays. I can only imagine what's in the new chinese boxes (more plastic).
Ya know. I'm still a novice player with just a few tunes under my belt, Back Door, J'ai Passe, Allons a Lafayette, and a few others, albeit without all the fancy ornaments in between the notes. But after knowing now what's in those cheap accordions, I can only desire to buy a new or relatively new used handmade accordion. An Acadian would be great because Marc's only 15 minutes away so help is easy to get to if I need an adjustment, etc.
The plastic "leathers" are called Ventile.
You will find them on most european accordions now a days.
They are very light and react faster than the leather checks, but they are not as good in humid climates because they do not "wick" the water away from the reeds like the "real" leathers and your reeds can end up "rusting". (this is bad)
But in California, Arizon and New Mexico, they work great.
You can glue them back in with a little "Crazy Glue".
Be very sure not to get the glue anywhere but on the very end of the Ventile or you will be sorry. :(
(Too much glue will also "attack" the plasic and make it curl so it does not seal correctly. This is worse then not having it on at all.)
Keeping your leathers in good condition will help to extend the life of your reeds and also make your accordion respond better.
Experimentation with such things must be done with very calm hands and sound mind. It's good you stressed that the very tippy bottom should have glue. I once screwed up a reed when glue leaked onto the reed, opposite. Luckily it wasn't much so I cleaned it out with fishing line.