Your comments on him recording it at home got me curious as that as how I remembered the story too...looks like yes it was but it was also mixed and remixed and engineered a few times before release...cool wiki
HERE
There is a bit more info on the album recording in this one:
McCartney (album) - Wikipedia
This is where the "4-track" thing hit home and stuck in my head for the next few years until I got my 4-track.
I considered the TEAC 4-track for awhile, but I ended up getting an Akai 4-track with basically the same features/capabilities, and then I added an Akai 2-track, after using a cassette deck for awhile to mix down to.
The Akai decks had great heads on them using crystal ferrite "glass" heads instead of metal heads, they didn't wear...and the cool (unplanned) thing with the two Akai decks, is I realized that the tracks lined up "perfectly".
IOW, I would record on the 4-track and then mix down to the 2-track, and then simply take the reel off the 2-track, and put it on the 4-track deck...and the tracks lined up!
Granted, nothing was properly calibrated or aligned (not that I knew at the time anything about tape deck calibrations anyway), but for me it just worked ...so I was able to then add 2 more tracks, skipping one bounce generation.
But yeah...it all "clicked" for me following that McCartney album and reading about what/how he did it...realizing that his basic rig was within reach.