As I understand it, the Four Class System was a simple ethnic hierarchy as follows:
Lane, May & Mote each claim that the former subjects of the Southern Song Dynasty constituted the Southern Chinese, while the rest of the Mongols 'Chinese' subjects constituted the Northern Chinese. However, it really makes no sense that the Yuan would simply divide the Chinese geographically, as the Four Class System was an ethnic hierarchy.
If the Yuan's aim was to portray China as divided, a geographical explanation for the terms Northern vs. Southern Chinese would make sense. From my readings, this doesn't seem to be the case - if anything, the Yuan seemed to be proud to control a unified China and derived some legitimacy from this.
Lastly, there would be fews ways to distinguish a Han from North China vs a Han from South China - their location and, perhaps, spoken dialect are all that come to mind.
So, was Northern Chinese simply a catch-all for Khitans, Jurchens, and Koreans and Southern Chinese a catch-all for the Han?
Two issues here. The first being that I think the difference between North and South is/was bigger than you're giving it credit for. The second being that the class system wasn't really a thing of such importance. First the first one:
However, it really makes no sense that the Yuan would simply divide the Chinese geographically, …
It's not that they were dividing the Chinese based solely geography. It was based on loyalty. The South was long a place of resistance against the Mongols, even well after the fall of the Southern Song court. Those who were citizens of the land held by the Southern Song were classified as Southern Chinese, and those of the northern areas which had previously fallen to the Jin (Jurchen) were Northern. However,
…as the Four Class System was an ethnic hierarchy.
you also need to think about the issue of ethnicity a little differently than just "Chinese" as a thing. Throughout Chinese history the Han were not the single unified group the PRC paints them to be today. There is significant genetic difference between northern Chinese and southern Chinese groups, with Southern groups showing a closer relation to Southeast Asian groups. This affects culture as well, and the North-South variation is a well-studied topic in genetics.
The point is that ethnic groups are and always have been flexible, and as much the result of self-identification as anything external. The borders and labels don't remain consistent over time, and it's entirely understandable that, while there would have been a common identification among the two groups with the huaxia category of Chinese culture, that's not to say they would have formed a single ethnic group.
Lastly, there would be fews ways to distinguish a Han from North China vs a Han from South China - their location and, perhaps, spoken dialect are all that come to mind.
There were major cultural and linguistic differences even in the 1200s, and ones which were entirely apparent at that time. Mandarin had already developed as distinct from its sister varieties by that time, and groups such as the Hakka were also socially distinct. It's precisely these differences which are the focus of my own research on the historical development of diversity within the Sinitic languages. The fall of Kaifeng and the migration to Lin'an mark a period of huge importance in the development of a number of Chinese languages, precisely because the difference between language and culture was so marked.
was Northern Chinese simply a catch-all for Khitans, Jurchens, and Koreans and Southern Chinese a catch-all for the Han?
Nope. And also keep in mind that Han as an ethnic group isn't really an eternal truth going back into history. Many groups that are today Han have had their role as Han disputed quite publicly in the past.
The Khitans and Jurchens would have been identified as Khitans and Jurchens. Same with the Koreans. The Northern Chinese were still at that time called Han, as distinct from the other groups mentioned.
However, then there's part two.
Yes, there were such categories, and yes, the official laws did reflect that, but the reality was quite different. The four classes were way less significant in any sort of limiting fashion than any other so-called caste systems I'm familiar with. Quoting from the Cambridge History of China (volume 6, around page 610) because it makes a good summary:
Various scholars over the past half-century have dispelled the notion that a sort of caste system was at work in Yuan China. It is quite easy, in fact, to find examples of Chinese holding official positions (such as darughachi) that according to government regulation they were not entitled to hold. The frequent exceptions to the Yuan regulations, which attempted to set aside certain offices for specific ethnic groups, indicate a fair degree of political mobility, although the paths to office were nonetheless not seen as traditional by Chinese scholars.
The written laws may have said one thing, but the reality for many was something quite different.
References
Twitchett, Denis C; Franke, Herbert; Fairbank, John K. (1994) The Cambridge History of China: Alien Regimes and Border States, 907-1368. vol. 6. Cambridge University Press
Hon-ming Yip. (1980) The class system of Yuan society: A critique of Meng Siming's Yuandat shehut jieji zhidu. Journal of Asian Culture. vol 4. pp 82-106