I'm fairly certain that Paul McCartney just made that episode up. Or got the details wrong.
If there were "boring" episodes--I didn't find them so, though I was in my early teens at the time and had a better attention span--they were only so in comparison to the way-out episodes featuring the rogue's gallery. There's one episode called "Paging the Crime Doctor", involving the mob doctor working for Rupert Thorne--I think he was actually his brother, but I can't remember--that flirts with mid-70s style TV storytelling in terms of the plot. But it also features a scene with the bad guys going after Batman with a stolen surgical laser. A fucking laser. And it was a more or less scientifically accurate portrayal of a laser, too, from what I remember--no dodging Star Wars-esque coloured bolts of light.
The great thing about this show was that it captured the "serious" "realistic" "grim and gritty" Batman--which, as much as we may groan at the terms, does fit nicely with the character when handled properly--but it blended it perfectly with the wild Silver Age weirdness of his rogue's gallery. This was embodied in the Joker, who they pretty much nailed in a way I don't think anyone had done before--he could be funny and over-the-top and imaginative and even campy, and still be fucking scary. You saw hints of this in the Bronze Age comics and Jack Nicholson's performance, but TAS was the first to get it exactly right. I like a lot of other versions of Batman, but usually they lean too far into camp (Adam West, Schumacher, the entire Silver Age) or into being "realistic" (The Dark Knight Returns, Nolan's movies).
Ironically, the only other one to avoid this was Burton, who put an emphasis on stylization to which TAS owes a big debt. This may be one of the reasons I still enjoy the Burton movies a lot, but it was key to the success of TAS--the fact that, despite the lack of truly wild SF shit, at least at first, this clearly was taking place in some heightened reality. So they could do relatively realistic stories about gang wars, or they could bring in the Riddler, and neither felt out of place. And as a budding artist, the stunning designs and compositions always made it worth rewatching, even the supposedly "boring" episodes Paul hates so much.
I will say, though, that by the time they revamped all the character designs and brought in Nightwing and Tim Drake, the show was pretty clearly losing steam. "Over the Edge" is great, but the rest of the shows from that era felt kind of rote, and it was too cluttered with supporting characters. And I hated some of the new designs, like the new Joker, who was way too simplified. (I notice they changed him back for his appearance in the Batman Beyond movie.) Of course, even some earlier episodes felt a bit like they didn't really know how to keep the pace going--both the Riddler and Mr. Freeze kick ass in their first appearances, but none of their subsequent appearances are particularly good. (Well, OK, the one with the Riddler drawing Batman into virtual reality was kinda neat, but even there you could feel a bit of flop sweat.)
Damn, I really did love that show. Gotta revisit it.
Personal favourite episode: The Man Who Killed Batman. Another episode where Batman's offscreen for most of it, but it's brilliantly done. The Joker's eulogy is classic. "Well that was fun. Who's for Chinese?"