It's almost certainly possible to bypass the adapter, run the longer wire of the new part down to whatever that ultimately connects to, and wire it up directly there (ripping out the internal wiring the old part connected to so there would be space to run the new wire through), but that's more refrigerator disassembly than I have any intention of doing.
I've left out the mounting screws in hopes that the adhesive strip is enough. The screws are tiny, human necks don't bend the right way to see where the holes are, and my hands are too big to feel those into position. The screws have been moved to a labeled bag in case that turns out to have been a horrible mistake.