(Disclaimer: Everything below applies to Korean spoken around Seoul area. I don't know the state of vowel length distinction in other areas.)
Well, the sentence you partially quoted already answers the question:
Although the "long syllable rule" is still taught in school in a standard curriculum for Korean language, vast majority of Koreans quickly forget the rule as soon as the final exam is over. So if you are a Korean language learner, there is no need to bother learning this rule.
I would paraphrase it as: Although the "long syllable rule" is still taught in school, the rule no longer holds, so most Koreans have to "learn" the rule by memorizing the list of long/short words, which they quickly forget once they're past the exam.
Contrast that with an actual sound distinction that native speakers still make. Can you imagine an English-speaking kid "forgetting" that "that" and "thin" has different "th" sound, once the exam is over? Of course not! There is an actual difference, and every native speaker makes the distinction, so you can't forget that any more than you can forget "did" is the past of "do".
In contrast, vowel length distinction is an obsolete rule in Seoul area; tones died even earlier. I think most people claiming to "subconsciously" making the distinction are just good at fooling themselves.
In conclusion: I don't think a language can be "a little bit" tonal. Either it is a tonal language, where speakers can distinguish two otherwise identical words based on tone alone, or it is not a tonal language, where speakers cannot make such a distinction. Modern Standard Korean clearly falls in the second basket.