I feel relieved now that I have everything put back together.
The trick? Well, it wasn’t so much a trick. I missed an important step in disassembling the Xbox, in the first place. The top actually detaches from the front! I didn’t catch on until I saw this video, the first I have seen actually demonstrate disassembling the front of the device. Now that I know this, I can probably do a complete drive swap in under an hour!