The other thing interesting about the Korean alphabet is that the symbols take "notches" also have the same mouth formation. The notch is an indication to use the same mouth shape, but to use a "harder" sound. So the G sound and K sound have a similar mouth shape, as well as the D sound and the T sound, the J and CH sounds.
My understanding, also, is that there are no exceptions to the rules, so it makes it very easy to read and write, which is awesome.
There are exceptions to those rules, hundreds of them. ㅊ(chieut/치읓) as the bottom consonant has a really difficult and varying set of rules for different words, different consonants before and different consonants as the start of the next word.
Take this word 청령리 (a large transit hub for intercity bus and train travel. The transliteration from that guide is Cheong-ryeong-ri which is completely wrong. It should really be transliterated closer to Cheon-nyeong-gi. If you say that word as that guide has shown you no one will have a clue where you want to go.
I live in Seoul. Non-Korean speakers frequently struggle to tell taxi drivers or to get directions to three different places: 시정 (sijeong - city hall), 신촌 (sinchon - a district in the north) and 신천 (sincheon - a district in the south). First, anyone who hasn't learned 시 is closer to 'shi' than 'see' gets nowhere. Then there is the subtle difference between ㅓ and ㅗ; ㅓ is more like 'audio' than 'eo' and ㅗ is a short 'o' sound. These things take practice. The ㅈ in 시정 becomes more of a 'ch' sound than a 'j' sound. The difference is about the same as the difference in pronunciation between the 'ch' at the start of church to the 'ch' at the end of it in English. Hard to notice, but there is a change in most peoples pronunciation for that. There are people who have been here for years who still get a taxi to the wrong place every now and then.
15 minutes might be enough time to learn the symbols if you are an adult who can already read other languages. A week is more realistic for actually remembering them. A month of practice and you will probably be able to read what people write on bits of paper at a speed that allows them to listen and follow what you say. Learning alphabets isn't too hard but it isn't easy either (I learnt the Mon alphabet in a day with a good teacher).
When I was young, I had a problem learning to differentiante between the 'ㅓ' and 'ㅗ' sounds.
My way of remembering was to think of 'ㅓ' as the weaker (i.e. less powerful) sound associated with "dumb" or "uh" (same pronunciation), and 'ㅗ' as the stronger sound associated with an exclamation of "oh!" (same as your short 'o' sound reference).
My understanding, also, is that there are no exceptions to the rules, so it makes it very easy to read and write, which is awesome.