You have a good grasp on the mainstream view, that the song was purely an attack on Paul.
However, when an artist creates something, then tells me "this is what I really meant," I do him the courtesy of trying to see how that might have played out.
If you look at some of the lyrics of the song without reference to the fact that they are titles of Paul's songs - admittedly, difficult to do, but try it - you really can interpret much of that song as an act of self-loathing.
"Sgt. Pepper took you by surprise" - Paul thought up Sgt. Pepper - how could it take HIM by surprise. It probably took Lennon by surprise, all right. That's when Lennon lost control of the band for good, the pivotal moment in Beatles history.
"The only thing you done was yesterday" - Lennon's post-Beatles career was nowhere near the level of success of Paul's. Paul was riding high, Lennon was drifting.
"and since you've gone you're just another day" - Paul led a successful band, while Lennon was just another solo artist.
Now, I admit that a few of the lines do seem like direct shots at Paul - the "pretty face" line, the "muzak to my ears" shot, the "you was dead" reference. I would argue that, if you take away the fact that they appear aimed at Paul, and instead see them as being aimed at a Beatle, you can also see them as an indirect attack on himself - former leader of the Beatles, the man who more than anyone created them and identified with them. One can just see John, angry at himself for putting out increasingly unappreciated songs, belittling them as "muzak," and thinking of himself as being just as dead inside creatively as the "freaks" said Paul was in reality. The idea of the Beatles as just being "pretty faces" whose time had passed may have preyed on his mind. I am reminded of a curious statement Linda said much later, that when John "retired," it wasn't because he didn't want to write - it was because he couldn't.
That's why I said it was oblique. Paul is never mentioned by name, and Lennon leaves it up to the listener to draw the appropriate conclusions. Ultimately, the listeners were completely fooled, the joke was on them (and Lennon was famous for doing stuff like that), since he by his own admission was talking as much about himself as his former bandmate in this song. Using the titles to Paul's songs was the perfect cover, camouflage for his real target to the tenth degree. Nobody understood that he was sad, depressed, angry at himself, and bitter about his own failings, both in failing to keep the Beatles together (which he could have done, quite easily, by getting rid of Allen Klein), and in going down a path that increasingly must have looked like it led nowhere.
It was so brilliant, so deceptive, that to this day, and despite his own explanation, listeners don't catch on to what he was doing.