It's interesting that if you don't click the Miyazaki article and just read the headline, you assume he's talking about the choice of stories or the subject matter of anime right now…when he's actually talking about the art and animation process. If the former, he'd be 100% right. But since it's the latter I think there's room to be contrarian.
I'm no artist, but I have noticed some significant artistic changes since 2006, and they're almost memetic in how quickly they've taken hold in popularity. For example:
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A big part of these changes derive from the advent of digital animation. What digital animation has done in simplifying the character art, it's made up for in more dynamic backgrounds and camera angles. Check
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HappinessCharge! PreCure, released just last week. You see a ridiculous 180 degree pan and zoom at the end there, all which a lot of Lucas-esque dense, busy movement. Having watched a lot of the classic OVAs in the past few years (like Lodoss, GR, etc.) I can say that digital technology has managed to simplify a lot of what animators in the '80s and '90s wished they could do, and invented techniques to try and attempt such things. Techniques like the "Itano Circus" are indeed going by the wayside, not unlike the sword skill of Western Europe, and while they're aesthetically beautiful it's more a matter of techniques evolving to meet the expectations of new technology. A lot of the conventions people like from anime, like the stereotypical sweat drops, movement lines and expressions, are based in part on efforts to simplify animation.
So, I can't really agree with what Miyazaki is saying. While certainly true that the mean anime looks like garbage compared to the mean anime of ten, twenty years ago, 1) there are more anime out then back then, and 2) anime on the same budgets as 10-20 years ago look way better with the tech of today. The garbage we see today is made on shoestrings by comparison, with low overhead to maximize the sharking prices out of the BD-releases. For example, Little Witch Academia looks amazing and the sequel is going to be OVA-quality 30 minute beast costing a paltry $150,000, but that's like three times what a normal TV anime episode would cost to make these days. And those shoestring anime look better than many TV anime from the late '90s that weren't done by a premier studio.
Is it sad that the Vladimir Tytla's of the animation world are going extinct? Yes. It's sad whenever a skill or style dies out when it's still appreciated. But I don't think animation is doomed just because of that. Nothing in the West has really lived up to 1940 Disney animation-wise - those works are triumphs of animation that went to a place no one else can follow - but we've done alright since.