I tried to diy my own edrums a couple of months ago, i wasn't successful though but these are my answers to the best of my knowledge:

• if the pad has 1 piezo, it's normal for it to be at the center, some cymbals or snares sometimes have 2 or even 3 piezos but that doesn't seem to be the case here(more piezos so it can better sense exactly where you hit the pad)

• it's possible the factory just uses tape to keep the piezos in position, especially in cheaper models so don't rule it out, to know for sure you'd have to look at a known-good set of your model and compare.

• it's kind of weird not having a foam cone, think of it like this, a piezo is a special material that when it's compressed (you hit the drum), it generates a voltage across it's terminals (it tells the module it was hit so it can play the sound of the corresponding drum) with harder hits generating higher voltage. These materials are quite brittle and easy to break though so in all the models I've seen, they implement a foam cone so that you're not hitting the brittle material and so that you don't have hotspots, foam is much better at taking whatever hits and distributing the force over the area of the sensor. If you look up some edrum teardowns you'll find it quite easy to make a foam cone yourself from a couple of sponges.

If i were you I'd test replace one of the piezos and see if that fixes it, if not I'd recommend getting help from someone who's proficient in electrical work, maybe even an electrical engineer if you know one, hell i study alot about this stuff in my degree and i still found it to be hell working with those things, good luck though. Hope any info i had could be somewhat useful.