Adjustment Screw Insert
I'm not sure what you mean by "insert." I don't see a particular insert in any of your photos. I can tell you that if your windows are drifting down, then you need all new balance springs, not just an insert that screws into the window.
Your particular shoe housing looks like our 15-087, which is designed to be used with the Series 180 spring. That's just an educated guess, though. Look these parts over and see what you think. The spring may be all you need.
I've never encountered a setup like that before. Not with a modern window, anyway. Some old windows from, like, the 1940's work by friction but no window company does that anymore, and especially not with the hardware you have now. What you're showing me in your first two photos is a pivot carrier for a coil balance spring. It's designed to carry the coil spring, which itself does all the lifting and holding as you operate your sash.
If your windows have no such springs, then to put it bluntly, someone messed up. You definitely don't want to just pop a screw in there to hold the windows up. I can't see how that won't just make matters worse. It's simply not how your window is meant to operate.
I'm not sure as to the extent of the changes made to your window when the installers did this, but it's definitely some kind of custom job. If I were you, I would see if you can get a hold of the installers directly to come out and fix their mistake. If you don't know who they are or can't find them, then you definitely want the help of a professional to undo what they did. You need proper balances installed in this window to get it functional again.