> This is because if, when finally screwing the parts together, the screw has to thread its way through, then the gaps between the parts won’t tighten when you tighten the screw, because there’s no travel between them.
That's why wood screws are only threaded partway up their length. Since these screws are self-tapping, they need to account for the problem you correctly identify here, where the screw can't clamp the parts together properly because they cannot move relative to one another as the screw is tightened into place. Ending the thread partway up the screw solves this by permitting the part nearer the screw head to move freely along the shaft, so the fastener can clamp them together properly as it's tightened.
The machine screws you're using, conversely, are threaded all the way up to the head because they are not designed for self-tapping, but rather to be used in holes which have had threads tapped into them. That's why they are giving you the trouble they are. Happily, there are a few different ways to make that trouble go away!
Making the entire screw path wide enough that the screw doesn't have to tap any threads, and securing it with a nut, is one option, and as you've seen, it can work. On the other hand, you lose something in clamping force, because there's less bearing surface, and you also have to choose between machining (or designing) a nut pocket into the part, or having the nut and the end of the bolt stick out past the profile of the case.
Another option, which replicates the benefit provided by the design of self-tapping screws, is to make the screw hole wider, but only in the part nearer the head, and leave the far part's hole narrow enough that the screw will tap threads into it on the first insertion. This way, you have the part near the head free to tighten, which solves the problem you saw with your first designs, and you get the maximum clamping force possible from the screw without having to tap threads for a machine screw - and, with a carefully chosen (or trimmed) length of screw, you can also have the path end in a blind hole in the far part, so that the only part of the screw visible in the finished item is its head. (If you care, of course.)
And a third option, naturally, would be to actually tap threads into your screw holes - but you need a tap set for that, and I'm not sure there's much point. You need taps to cut threads into metal, because metal is pretty hard, and you need something very hard to cut it cleanly so that your screws, which are only pretty hard themselves, aren't ruined when you try to use them. Plastic, on the other hand, is very soft, so you're not actually hurting anything by letting your screws self-tap. You just need to adjust your design to account for it.
Hope this helps!