And to what extent can the Hakka's historical position within their broader ecumene be compared to other minority groups worldwide - most especially European Jews? Both are groups who have been severely persecuted over history, but nonetheless have played a disproportionate role in politics, business, and thought, both within Europe/China and in their respective diasporas?
When discussing this topic, it probably makes sense to start by discussing who exactly the Hakka people are. This is particularly important because there is significant ambiguity around the term.
The Han Chinese are generally considered the largest ethnic group in humanity. But as you can probably imagine for a group consisting of over 1 billion people, Han are not perfectly homogenous but can be subdivided into cultural and linguistic subgroups.
The term Hakka (in Chinese 客家 - literally, guest families) originated as an umbrella term for Han Chinese that migrated internally within China, typically from north to south. Because China's census usually only tracked Han vs. non-Han populations, and the major migrations occurred during periods of major turmoil (e.g. the fall of the Ming dynasty to the Manchu Qing), it is hard to precisely track the migrations of Hakka people. For the same reason, it is also hard to be sure if historical references to the Hakka are describing a continuous lineage of people, or if the same term was used an umbrella term to describe distinct waves of migration that occurred even as earlier waves assimilated.
More recent genetic and linguistic evidence broadly supports this understanding but does not really shed much more detail. Genetic studies indicate that the Hakka have slightly more central Chinese ancestry compared to native south Chinese groups such as the Cantonese, but ultimately the Hakka and Cantonese are more closely related to each other than either are to north or central Chinese. Linguistically, the Hakka language is most closely related to the Gan language, which is the Han Chinese sub-language spoken directly to the north of Hakka speaking areas.
It's hard to quantify the claim that Hakka have played an outsized role in modern Chinese history, so it's hard to say for sure whether the premise of this question is valid. But I do think some of what we observe in Hakka history seems easier to understand in that context.
A significant share of the Hakka people appear to have come to southern China fleeing the Manchu Qing conquest of China. In that sense, while it is hard to prove either way it may not be coincidental that many of the Hakka settled in coastal regions that were hotbeds of anti-Qing sentiment and that many Hakka supported or even led anti-Qing movements like the Taiping rebellion, and in turn the Qing dynasty retaliated against the Hakka community following the Taiping rebellion.
Likewise, contemporary sources depict the Punti-Hakka clan wars as a clear-cut conflict over land (the term Punti/本地 literally means "native") in which local Cantonese people violently evicted people that they saw as migrant interlopers.
In this context, I'm also not sure I would see this as very comparable to the situation of the European Jews. The Jews are a minority with a long history and a clearly defined identity centered around their distinct religion. By contrast, the Hakka were less clearly an "other" in Chinese history - the modern Hakka only emerge as a recognizable group in the past few hundred years and it is unclear whether or not they can be connected to earlier groups using the same label.