Not OP, but I agree with you up to a point.

What George and John couldn't do, though, was write songs that were both deeply personal and universal; an art that Paul mastered and understood better than anyone.

I think both were capable, but certainly not with the same consistency as Paul. But they didn't seem much to care to, though, either, after a while, while Paul still did.

Paul's music is melodically brilliant, layered with exquisite harmonies, stylistically varied, thematically cohesive, and (often self-) produced with a tonally coherent and goal-oriented vision, while somehow being experimental yet accessible and personal yet universal. I think Paul's affability and optimism (and yes, a string of lazier or intentionally cheesy singles), for various reasons, led to his music not being understood as "serious" and "deep", when in fact, he is quite possibly the greatest recording artist ever.

I think, though, that his not being taken seriously or seen as "deep" is largely his own fault and directly a result of how he started out his solo recording career, and continued to do on many later releases as well. McCartney was an intentionally underproduced album, which was followed up by the super cheesy "Another Day" and the B-side "Oh Woman Oh Why" with the pretty lazy lyrics. Ram was unfairly criticized, but then this was followed up with the singles "Mary Had A Little Lamb" and "Hi Hi Hi". More super lazy lyrics. Wings then debuted with Wild Life which starts off with "Mumbo" which barely has lyrics, and "Bip Bop" which has really lazy lyrics, and then "Love Is Strange" which is a cover.

Red Rose Speedway came next which was called more schmaltz by Paul, which it is. Paul finally got good reviews with Band On the Run and Venus and Mars and deservedly so. But those came after many, many releases of under-produced or schmaltzy recordings, and then after those two albums, he went right back to that style with "Silly Love Songs" and Speed of Sound and continued in that same mediocre soft rock vein for the rest of the 1970s.

So I think it's partly Paul's own fault. It's easy to look back at Paul's body of work in retrospect and pick out that 30-40% of it that is genuinely both really good and really thoughtful in the lyrics department. But a good chunk of the rest, especially on some of Paul's early albums, feels very tossed off or else saccharine and too mass market. In chronological order, as it happened, it's much easier to see why he's been judged the way he has been.

To me, I think John was consistently the better songwriter during the 70s, but that it didn't necessarily translate into more better albums. His songs were generally good, but as the 70s wore on, they tended to be ruined by bad production more than anything else (though Sometime In New York City was arguably ruined by political lyrics as well).

If you ever listen to the "Let It Be" sessions, you can see how this all materialized through the way they worked together by the end of the 60s. John and George tended to have a song all written on guitar or piano, with all the lyrics finished (though occasionally asking for help with a line here or there), but no real idea on how it should be played with the rest of the band. They'd then work together suggesting different arrangements and intros and whatnot, until it was finished.

Paul worked almost the opposite way. He had the tune done and usually the chorus, and maybe the first verse, but not the whole lyrics. Instead, he had ideas of how the tune should be played with the rest of the band, or with an orchestra, or however it sounded. And he'd show up with a verse done and throw it out to the rest of the band for ideas on the rest. On "Let It Be", he did it for the songs "Let It Be" and "Two of Us" and appealed to John for help with the middle of "I've Got A Feeling" which John wrote, while "Get Back" was developed as a jam in the studio, with the chorus initially an ad-lib and all. The only song whose lyrics Paul had finished for that album before coming to the sessions were "The Long and Winding Road".

So Paul's solo material tended to have really great production, but the lyrics often left something to be desired because Paul never had the confidence or patience to finish them properly. John often had great lyrics with a tune that had a great hook, but it would end up ruined by layers of horns and saxes and other 1970s production techniques because he usually deferred to producers or other staff other than himself. And George was kind of ruined by both--as you said, often sanctimonious lyrics and bad production because he'd never got a full handle on either.

I think if Paul had been much more confident/patient/thoughtful with his lyrics, and been more patient with some of his productions, you're right, he would quite possibly deserve to be called the greatest recording artist ever based on his work with the Beatles and his solo work through McCartney II. But I also think if John had been much more confident in his rock and roll vision in the studio, and recorded all the exact same songs with the exact same lyrics but just played every album like Plastic Ono Band, or produced them like Neil Young produced his albums during the 1970s or something, then we'd also be talking about John as the greatest rock and roll star of all time, too, based on the songs he wrote through Double Fantasy.

But the fact of the matter is, neither is true, they both fell into traps that ruined some of their albums and/or singles, and so there is more than enough mediocre material mixed in with the great that they fall short of "greatest of all time" as solo artists. Paul at his best is easily as great as anybody else, but he certainly recorded plenty of material where he deserves the "shallow" and "not serious" labels. Same with the criticism of John. He was a better rock and roller than a pop star, but he rivaled any of his peers in rock or pop when he was at his solo best, but he also deserves the labels of "too political", "too much Yoko", and "badly produced" for much of his solo work at the same time.

It's really why the Beatles worked so well together. John and Paul had a great working relationship where they could eliminate all the bad stuff and only record the good stuff, with the best lyrics and with the best musical arrangement that the two of them could come up with together. Without that partner, nobody else around either of them proved to be a reliable partner to get the best work possible out of them individually.