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.