I don't think the Chinese cultural tradition that had gone for centuries was really connected to how they were thought of in the 20th century. Japan generally had the view that while some of their traditions might have originated from China, it had since then degenerated and now Japan was the rightful heir of the "real" tradition they had built on top of the original. I mean, they had been using those characters for hundreds of years already by the 20th century, it is much more likely they saw them as their own rather than as "Chinese". Therefore I don't believe there is any contradiction between seeing the Chinese as inferior in the 20th century while using the characters.

In a general sense, why did the others eradicate the characters while Japan did not? I'm not sure, but it seems to be dependent on each country. I think the central process here was that to promote literacy, every Asian country got rid of the very difficult Classical Chinese as the main literary language in the early 20th century and replaced it with something closer to their local spoken language. In Vietnam, Latin script was introduced by the colonial French administration while in Korea the Hangul script was promoted by nationalists and suppressed by the Japanese so it gained nationalist symbolism. Still, South Korea only really gave up the Chinese characters fully in the 1990s while the more anti-Japanese North Korea abolished them in late 1940s. So, the reason for keeping or not keeping Chinese characters was mostly a political issue unique to each country.

I've never heard of any major attempts in Japan to eradicate the kanji entirely, though I wouldn't be surprised if there were some arguments in the margins for switch to a Latin script or some other form of simplification - these were debated in pretty much every country during their modernisation, as even some Chinese argued for abolishing the characters in favour of some kind of alphabet.

TL;DR: Anti-Chinese sentiment during WW2 and usage of Chinese characters in Japan are not contradictory, there is no connection between the topics.