TL;DR - Sino-Korean, as documented, never used those 치두음/정치음 hangeul jamo.
By the time of Sejong the Great in 1440s, the form of Middle Chinese that was studied (and thus different to the nascent Beijing 'Mandarin' 官話 관화, sometimes called the 公談 공담, of the Ming dynasty) had a 32-initial system.
This differed from the Middle Chinese 36-initial system of the 韻鏡 Yùnjìng 운경 rime table, for which we have Song dynasty manuscripts of the late 12th century, and whose system of Chinese likely reflects the Chinese of the Tang dynasty. The 'lost 4' is composed of the 舌上音 (설상음) series 知/徹/澄/娘, believed to be palatal coronals /tʲ, tʲʰ, dʲ, nʲ/, sometimes transcribed in modern times as /tr-, trh-, drh-, nr-/, which hints at how they (generally) merged into the series that ended up as retroflex consonants in modern Mandarin.
Note that these 36 differ substantially from the 37 initials of the Early Middle Chinese of the Sui dynasty represented by the 切韻 Qièyùn. From Early Middle Chinese to Late Middle Chinese, there has been the merger of the palatal affricate series 章/昌/禪/書/船 with the retroflex affricate series 莊/初/崇/生/俟 to form a new merged 正齒音 (정치음) palatal/retroflex series 照/穿/牀/審/禪 (note by the way how 禪 has de-affricated too) has occurred, as well as a split of the labials, forming a newly separate 非/敷/奉/微 labial fricative series. Thus a net change of -5 + 4 = -1.
Thus, that leaves the following three 齒 'sibilant/affricate' series, which in the 훈민정음언해 Hunmingjeong'eum Eonhae 訓民正音諺解 are:
- 齒頭音 (치두음), alveolar non-palatalised affricates 精/清/從/心/邪, which were given the long-left leg spellings in hangeul: ᅎ/ᅔ/ᅏ/ᄼ/ᄽ respectively.
- 正齒音 (정치음), retroflex or palatal affricates 照/穿/牀/審/禪, which were given the long-right leg spellings in hangeul: ᅐ/ᅕ/ᅑ/ᄾ/ᄿ. This series is the one that would have taken in the former 舌上音 (설상음) series as a whole, but not necessarily all of them.
- 半齒音 (반치음), some sort of voiced obstruent rather than a sonorant (as nasals and liquids would be), with the hanja 日. This was given the letter ㅿ in hangeul.
However, the main Chinese rime dictionary published alongside the 訓民正音 훈민정음 Hunminjeong'eum around that time, the 1448 東國正韻 동국정운 Dongguk jeong'un, used a 23-initial system in its Chinese annotation, merging 치두음 and 정치음 into one 齒音 치음 series 卽/侵/慈/戊/邪, and losing the the majority of the labial fricative series 敷/奉/微 ㆄ/ㅹ/ㅱ, as well as 喩 ㅇ as an initial, giving a loss of 9 initials.
Thus, as Lee & Ramsey summarise on pp.126-7 of their 2011 tome "A History of the Korean Language":
These were prescriptive pronunciations intended to “correct” the readings of
Chinese characters then in use in Korea. [...] But these artificial pronunciations were not, strictly speaking, simply imitations of Chinese. [...] In other words, the Tongguk chŏngun system was a theoretical construct representing a compromise between the Chinese rime tables and dictionaries and the Sino-Korean readings actually used in Korea.
However, we know that iotation was contrastive in Late Middle Korean, even for nativised Chinese compounds. Thus 쵸〮 for "candle", nativised from Chinese 燭 (modern Sino-Korean 촉) is iotated, whereas 초 "vinegar" from Chinese 醋 is not. These do not appear to correlate to whether the Middle Chinese or Old Mandarin characters were in the retroflex series or not. Palatalisation of the dental stops and affricates in central dialects of Korean is no longer contrastive before the end of Late Middle Korean, by the 18th century, although some northwestern Korean dialects still retain the distinction in some lexemes.
From an analysis of the late 19th-century Korean-language Early Mandarin Chinese textbook 你呢貴姓, we see that retroflex consonants in northern Mandarin were noticed, but transcribed using:
‘ㅣ[i]’ inserted between the onset consonant and the main vowel
This is also restricted by the limits of iotation in the script. In the 1883 화음계몽언해 Hwa'eum gyemong eonhae 華音啓蒙諺解, we thus see 這 is given 져 and 說 is given the hangeul spelling ᄉᆑ (yep - that's [슈ㅕ] in one syllable block), but 是 is 스, and 所 is 소 (rather than the expected 숴, to rhyme with 過 궈).
The use of iotation as a proxy for retroflex consonants fails when you have minimal pairs/groups like 桑 (Middle Chinese: sang, alveolar with no glide), 相 (Middle Chinese: sjang, with an alveolar initial and palatal glide) and 商 (Middle Chinese: syang, with a palatalised initial becoming retroflex later), all of which remain robustly distinct from Early Middle Chinese to Modern Standard Mandarin (sāng/xiāng/shāng, respectively).
