Every time period has its own unique context, so there could be many historical reasons or even coincidences specific to each 'unification' event that led to the north dominating the south. But I think there is indeed a kind of demographic and geographical trend that could be used to explain some of these apparent dominance.

Demographics

During the Han Dynasty, major of the population (as per census recorded in the Book of Han) lives in the plains surrounding the Yellow river, particular in today's Henan, Hebei and Shandong provinces, with population densities 10 times or more of the regions south of Yangtze.

Following wars and conflicts at the end of Eastern Han, the Three Kingdoms era, 16 kingdoms and North & South dynasties, and even climatic influences^(1), shifts in populations south of the Yangtze began to occur, but it was only until the Tang Dynasty that the balance shifted -- During the Sui Dynasty just before Tang, the northern provinces account for ~75% of the total population, but by the Tianbao (天寶, 742-756) era of Tang, following recovery from Post-Sui conflicts, the proportions became roughly even.

Following the An Lushan Rebellion that hit North China the hardest, and more conflicts at the end of Tang, by Northern Song (960-1127) we see a 41:59 divide, with 59% of the total population in the south.

So while it is true that lands south of Yangtze did came to support large populations comparable or even exceeding the north, it really only took hold nearing the past 1000 years or so. If we start from Northern Song, the unifications would be: Yuan (North), Ming (South), Qing (North), KMT (South). And that puts us at 2:2.

*Relevant data from respective books of each dynasty and 中國人口史 (Demographic history of China)

1. See for example: Climate Change and Migrations of People during the Jin Dynasty

Geography

While the south indeed has more hills/mountains that would make invading difficult, the reverse is actually true as well -- it is also difficult for those within the mountains to go on an expedition out of the mountains.

A famous example many might relate to would be Zhuge Liang's various northern expeditions through the mountains north of Yi Province (Sichuan today). Zhuge Liang was only aiming to obtain lands immediate to Yi up north (The Liang provinces and perhaps Chang'an), but his efforts mostly did not pay off, in part because of the sheer difficulty of terrain. When we look at descriptions of Zhuge Liang's northern conquests in Records of the Three Kingdoms, mentions of grain supplies were very common, such as:

冬.....圍陳倉，曹真拒之，亮糧盡而還。

Winter, [Zhuge Liang] sieged Chencang, Cao Zhen resisted [him], Liang's grains ran out and returned.

The paths out of the Yi province mountains were so difficult that the famous poet Li Bai once wrote:

蜀道之難，難於上青天！

The Shu roads' difficulty, [is] higher than that of going up the sky/heavens!

Additionally, in the south, many grain producing areas are small plains surrounded by mountains, compared to big wide plains up north. So even though the terrain affects both sides equally, the cost to bear is certainly higher for the smaller one, especially when they are on the offense -- the defending side can just hold out until the invaders run out of grains, as we can see in many of Zhuge Liang's conquests.

For the north, are the presumably less defensible plains necessarily to the south's advantage? Not necessarily. As Wu general Lu Meng puts it:

徐土守兵，聞不足言，往自可克。然地勢陸通，驍騎所騁，至尊今日得徐州，操後旬必來爭，雖以七八萬人守之，猶當懷憂。

Xu territory’s defending soldiers, are not worth discussing, and if we [go] can be overcome. However the land terrain is traversable, strong cavalry can gallop, and if you the utmost honored today obtain Xu province, [Cao] Cao later will surely come to contest it, and even if [we have] seventy to eighty thousand men to defend it, it would still be worrying

The exact same advantage for obtaining the plains can be a problem when defending, especially since supplies lines might be stretched for the southern armies going into northern plains!

So the geography is actually not that advantageous to any particular side in the grand scheme of things, and more about how either side utilizes the goods and bads of what their local geography has.

Geography & Demographics

Geography is also quite connected with demographics in some aspects. In the south, for quite a substantial period of time, the population centers are primarily located in the immediate plains/regions along the Yangtze, as compared to the more hilly/mountainous areas further south.

During the Western Han, commanderies further south from the Yangtze, such as in modern day's Fujian, Guizhou, Guangdong and Guangxi Provinces, have extremely low population. For example, Yulin Commandery, near today's Guangxi province, only had ~12000 households, while Danyang Commandery, immediately south of Yangtze reaching modern day Nanjing, had ~100k households for a smaller area.

During the Tang's Tianbao, Jiangnan Circuit, encompassing most of the areas immediately south of Yangtze, had 1.8m households, whereas Lingnan Circuit, encompassing most of Guangdong, Guangxi, Fujian and other further south regions today, only had 336k households.

In essence, by looking at it from a pure population perspective, once the northern armies cross the Yangtze and control the immediate south regions, they have quite substantial control of the region; and the rest is comparatively just ironing out the final details. But for the southern armies, the various plains and commanderies up north all had considerable sizes, and simply crossing the Yangtze and controlling immediate north regions would not grant the same degree of control as crossing into the immediate south.

In fact, a common strategy during times of divide was to relocate residents in the immediate north areas of Yangtze to further north areas, such that invading southern armies could not capture much (human) resources even if they do obtain these lands.

初，曹公恐江濵郡縣為權所略，徵令內移。

Originally, Duke Cao was afraid that [Yangtze] riverside commanderies could be taken by [Sun] Quan, [hence] ordered movement inwards [of residents]

But for the south, given the mountainous and hilly terrain, relocation can be considerably more difficult.

Hence, offenses made from the south often needs to go on longer and deeper into the northern lands to achieve control and unification, which often makes them more prone to a variety of issues like supplies, backups and so on as time passes.

All in all, this is an extremely brief summary of demographic changes and local geography that might influence the results. Though again, I must stress that while these factors do influence the final outcome, the specific contexts and decisions people of their time make have considerable influences on the final outcome that should not be left out of the equation.

Thanks to /u/10thousand_stars for writing a very broad-ranging answer (I was contemplating doing one myself but my expertise simply doesn't range that far). I do, however, want to raise a bit of an issue with the question's discussion of northern 'barbarian' polities.

To put it quite bluntly, the Mongols were the only nomadic people that successfully conquered all of China proper, and only did so thanks to an incredible amount of deliberate effort and diplomatic manoeuvring. The rivers running West-East through China, most prominently the Yangtze, were quite substantial obstacles to the Mongols, and would be circumvented through invading the indigenous kingdom of Dali in what is now Yunnan via Tibet and the more fordable upper reaches of the Yangtze in Sichuan. Even then, it took over 25 years after the subjugation of the Dali kingdom for the Mongols to overrun the territory of Southern Song. In other words the Mongol conquest in this region was made possibly only by creating an entirely new invasion route from scratch, on the part of a power with substantial enough control over neighbouring regions to be able to do so.

Earlier regimes of nomadic origin were generally unsuccessful in penetrating outside of northern China, as southern states were generally successful at maintaining stalemate lines on major rivers. For instance, northern Wei, which existed roughly from the late fourth to early sixth centuries CE, halted at the Yangtze; the Jurchen Jin state, which existed from the early twelfth century until its conquest by the Mongols over the course of 1211-34, extended only as far as the Huai. No nomadic polity before the Mongols established any sustained dominion of a region south of the Huai, let alone the Yangtze, and no nomadic polity would establish substantial control over any part of China proper again. The Manchus, it is worth noting, were not nomadic, outside of the northernmost portions of Jurchen-speakers known to the Ming as Yeren ('wild people'). The vast majority of the Jurchen people that became the Manchus were sedentary agriculturalists, with the patriarchs of free Jurchen households also involved in communal hunts to supplement agrarian food sources (in addition to fishing and sedentary herding where these were possible). While the Manchus and Mongols had a number of cultural similarities, societally speaking they were quite distinct.

All this to say that there is a tendency for discussions of the relationship between China and the nomadic world to fall into two opposite extremes: on the one hand there's the 'Sinicisation' model that asserts that nomadic influence was invariably ephemeral and that Chinese culture simply overwhelmed that of nomadic societies, and on the other there's the sort of hot take meme that nomads overran imperial China on a regular basis. The answer really lies somewhere in between: northern China was historically quite tied into the nomadic world, whereas southern China was not.

Thanks for this really great answer. I was wondering about the population growth south of the Yangtze so the details here are hugely helpful to my understanding.

Also great point on Ming and KMT coming after more population established in the south

I'm still reading the details of earlier dynasties so forgot about the origins of these regimes