A few points to start off:
Characters are not made up of radicals (which are strictly dictionary section headers); elements of characters which give meaning are called semantic components.
The phonetic part of 「漢」 is currently written as 「⿱廿⿻口夫」.
For brevity, this character will be referred to as 「𦰩」, but note that 「𦰩」 is actually a Japanese shape, with 「艹」 and not 「廿」 on the top, and writing it this way is incorrect in Chinese.
Chinese Etymology is wrong - there is no 「堇」 in 「漢」, and no 「難」 in 「漢」. This statement probably comes from older literature now known to be incorrect; both 「堇」 and 「難」 contain phonetic 「𦰩」, but most older dictionaries do not contain a definition of 「𦰩」, and so may make up character composition definitions based on circular logic.
「難」 (Baxter-Sagart OC: /*nˤar/) did not originally sound like 「南」 (/*nˤ[ə]m/), so the name Nanzheng (南鄭) is not relevant.
Meaning of 「漢」
「漢」 indeed originally was the name of a river. From the Spring and Autumn Period artefact 《敬事天王鐘》（《殷周金文集成》７３、７４）:
隹（唯）王㱏（正）月初吉𠩖（庚）𢑚（申）// (...) 𦣼（自）𠆦（作）永（詠）命（鈴），𢍌（其）// ⿵⿶⿱𦥑冖月頁（眉）𦓆（壽）𣞤（無）疆，𢿩（敬）事天 // 王，𦤴（至）于父⿰兄⿱𡳿土（兄），㠯（以）樂𠱭（君）// 子，江漢𡳿（之）陰陽，// 百歲𡳿外，㠯𡳿大行。
'Tis the day of gēngshēn of the first week of the first month of the King's reign*, (...)† self-cast this ceremonial bell. Wishing for boundless longevity, respectfully serving the King and the Elders, letting the noblemen rejoice, across both sides of the Yangtze and Hàn Rivers; and beyond one's years, with this bell, setting off on the Grand Journey‡.
*This timekeeping record isn't necessarily from the start of the King's reign.
†Inscription is faded out; this is expected to be someone's name.
‡Refers to the journey to/in the afterlife (death).
「漢」 was not originally associated with Chinese civilisation or ethnicity; rather, the first emperor of the Hàn Dynasty decided to name the dynasty after the river, simultaneously causing the entire unified nation to adopt the name of the 「漢」 ethnic group which originally resided around the river. Chinese civilisation, if it could be said to have a beginning, would have originated around Zhōngyuán (中原) rather than the Hàn River, and it is 「中原」 which gave rise to the name 「中或」 (central regions) >「中國」 (Middle Kingdom).
As one of the most prosperous dynasties in Chinese history, the adoption of the name 「漢」 eventually caused most long-standing notions of tribal boundaries and ethnic groups which have persisted all the way from the Shāng Dynasty to fade away, such that 「漢」 is now synonymous with Chinese.
Phonetic component of 「漢」
「𦰩」 is made up of 「黑」 (picture of a person with criminal facial tattoos) with the addition of an exaggeratedly-drawn mouth 「口」 on top, depicting a sighing criminal, indicating related meanings of bearing hardship, suffering (now written 艱、難), to sigh (e.g. out of grief, lamentation, suffering, or desperation) (now written 嘆), and punishment (e.g. 熯, punishment by fire).
The bottom portion of 「𦰩」 was often corrupted into 「火」 or 「炎」, leading to extra marks in today's shape which causes the bottom of 「𦰩」 to look like 「夫」.
In addition to its meaning role, 「黑」 may also serve as a phonetic hint for 「𦰩」, but generally, characters should only be decomposed at a surface level, and so it is inappropriate to decompose 「𦰩」 to see what kind of hints the resulting components provide to 「漢」.
Please note that 「𦰩」 merely plays a phonetic role in 「漢」, not a semantic (meaning) role.