I’m a native english speaker and I view “hate” and “can’t stand” as being synonyms. I can’t for the life of me draw a distinction between what either phrase means.
FWIW I thought "hate" and wrote "can't stand" because I intentionally wanted to tone it down just a little bit. "Hate" is a pretty biting word and I'm trying to use it less hyperbolically. The distinction is more one of politeness than anything in this case, so I agree with you.