The low literacy rates in pre-Hangul Korea had less to do with phonological vs. logographic writing systems and more to do with the fact that in Korea at the time Chinese played the role Latin played in medieval Europe: it was the language of scholars and officials, but most ordinary people couldn't read it because, well, they didn't speak Latin. The same thing happened with Hanja in Korea. When you're trying to get people literate in a writing system designed for a language they don't use in daily life, you're fighting an uphill battle from the start.
What is true that unlike the Latin alphabet, which European languages could adopt/adapt for their own use (or think the way Cyrillic was adapted for Mongolian), Chinese characters, being logographic, couldn't simply be repurposed to represent sounds of the Korean tongue — that's why Hangul had to be invented from scratch. That's one important difference between phonological and logographic writing systems, but it has little do with the question which system is better at spreading literacy.
This is anectotal but I've a friend who's well educated senior chip design engineer who's mother tongue Chinese but cannot read the written words since he has no background in Chinese vernacular education where people properly learn the multitudes of Chinese characters.
I know of at least one other instance. But note that the same is true of English speakers. Becoming literate in English isn't just a matter of learning the alphabet (otherwise a Frenchman automatically qualifies as literate in English simply because he knows the English alphabet). The way literacy is typically assessed is by asking you to read a typical paragraph from say a newspaper and then testing you for reading comprehension. Someone whose native tongue is English, but who for whatever reason has never had the primary/secondary education experience typical in an English speaking country is going to encounter the same sort of difficulty becoming literate in English as you friend in becoming literate in Chinese.
By his own account, it took this man [1] 7 years (during his late 40s/early 50s) achieving English literacy, and that happened in a English speaking country where he encounters English texts on a daily basis (I assume your friend lives & works in the West).