> Or is it that learning to read Chinese actually is a big problem in China?

Historically it was. Reforming the writing system (potentially even ditching it entirely in favor of a Latin/etc derived script) to improve literacy rates was a major topic among Chinese intellectuals during the 20th century.

Some combination of character simplification, reading and writing the vernacular instead of "Classical Chinese", brute force, and modern technology has made this less acute. But it still is not unusual for even educated native Chinese speakers to simply not remember how to write some uncommon character. (You will see this in English occasionally too, of course. I have to think twice when I write rendezvous.)