Anime from the last twenty years seems less directly influenced by foreign tendencies. Of course, since Japan is impregnated of the American culture (including the language), its anime cannot ignore this context. But decades ago, the Japanese pioneers like Tezuka were following the steps of previous pioneers like Disney, but now, the anime has grown enough to claim a relative independence, meaning their internal tendencies are much stronger than foreign influences.
> Miyazaki (who mostly made very Disney-like westernised things
The level of "Disney-like thing" hugely varies across Miyazaki's films. Mononoke is obviously Disney/Hollywod compatible, but Totoro not so much.
Miyazaki and Takahata, who co-founded Ghibli, repeatedly stated that the work that most influenced them was "Le roi et l'oiseau" (Grimault & Prévert, 1953, 1980). This masterpiece is of course a "western thing", but very far from "Disney-like", especially toward the second half of the film. I heard that the Ghibli museum often had screenings of it.