Greetings! We have before us one of those rather curious "what-ifs" of history. Namely, where there wasn't a "Scramble for China" as there was a "Scramble for Africa". This response will not go too far into the whole backstory of the two Opium Wars, the First Sino-Japanese War and the Boxer Rebellion, but instead it shall touch on the various problems facing European leaders when it came to the idea of "partitioning China". Note that this view is mostly taken from British-centric sources, so hopefully other contributors will weigh in with the various considerations from the other European powers and also Japan or America, as they were rather invested in China and its future at the turn of the twentieth century. Let's begin.

Note that I am writing as if though we were the various diplomats and foreign policy architects of the various powers in China*. It helps to "set the scene", so I apologise for the improper tense usage at times.*

Firstly, a major problem that we must consider in "Why the European's didn't go further in China" is a rather complex one which reveals a lot about the intricacies and nuances (or lack thereof) with imposing colonial rule: existing power structures. A massive problem facing any of the Western powers and Japan when it came to dividing up China would be how to overcome local resistance and impose their own version of the imperial system upon a country which had such a long history of autocratic and bureaucratic rule. For the British, this problem was particularly pressing, as prior to the Opium Wars many academics and politicians had looked upon China as a model power in the Asian region, whose "enlightened scholar mandarin bureaucracy" was to them an ideal system of governance for such a large and diverse nation. Granted, the external conflicts in the latter half of the 19th century and various internal schisms had seriously compromised the central authority of the Qing Emperor from Peking, but the matter of fact still stood: the Chinese populace far outnumbered any possible government representatives or offices the various powers could bring in to oversee newly colonised Chinese territories. Yes the British had experienced similar experiences with their colony of the British Raj, but keep in mind that prior to the British East India Company establishing a hold over the subcontinent, there had been various princely states, kingdoms, and two major empires (the Maratha and Mughal) which were divided against each other. In China, the Qing government had no official rivals within their territory (though there were several power-seeking parties and fledgling groups), and thus no major rivals to concern themselves with (or more importantly, no 'third-party' for the Europeans to ally with and colonise China with). Governing a nation with such a vast populace and a centralised (if increasingly powerless and obsolete) system already in place would be a legislative and administrative nightmare for London, Berlin, Tokyo, Paris, St. Petersburg, and Washington.

Secondly, we also have to consider the fact that beyond the local opposition any colonial efforts would face, there was also the foreign opposition. Remember, there was no one European power with dominance in China. All of them possessed "concession ports" and had signed "concession treaties" (or "unequal treaties" as later academia and public opinion would call them) with the Qing government, and thus each of them had interests in various parts of China. The British from their island holdings in Hong Kong wished to expand their influence into the Yangtze River valley, the French eyed the southern provinces from their colony in French Indochina, the Russians a foothold in Manchuria from their Siberian territory, the Germans the province of Shandong (and the key port of Tsingtao, or Qingdao if we use the traditional Chinese name), and Japan their share of interests from northeastern China. In short, if one European power took the initiative in scrambling for China, then all of them would follow suit. Likewise, if one was allowed to control large swathes of China's territory (and thus be allowed to capitalise on its economic resources), then it was a major fear in the eyes of European statesmen that another "Concert of Europe" would need to be imposed upon Asia. Under such a system, a fragile balance of power would have to be maintained, with no single empire holding disproportionately more control over China than the others. Such a geopolitical system had been difficult enough to maintain on the European continent, having to do so in Asia, with Japan and the United States as two additional parties, was not a task the leaders of Europe were willing to divert serious time and effort to. The Reverend (and historian) Gilbert Reid illustrates this complexity rather well, in his writings from 1900 (when the partition was very much a live topic of discussion) he states for one of the major European players:

"Russia knows that if she advances into China, other powers will do the same. The subjection of the whole of China to Russia is a very different thing to the complex partition of China. The former is impossible; the latter to Russian eyes is undesirable."

Directly linked to this concern of other powers exerting control over China, there is the economic consideration. As previously states, all of the nations concerned with China at the beginning of the twentieth century have their footholds, bridgeheads, treaty ports, and even embassies already in the 'Celestial Empire'. If a hostile European power were to attempt to, by force of arms, demand inland provinces and territory from the Qing government, it would likely jeopardise (or at the very least have some knockback effect) on various economic interests that other companies belonging to other nations possessed. Case in point: the French. They have "settlements" in Shanghai and Tientsin, a "concession" in Hankow, and major political influence (in the form of an embassy) in the capital of Peking. Her companies are the largest investors in the railway being constructed between Hankow and Peking. To see another (rival) power take control of China's complex and bureaucratic economic system by force would have serious consequences for French economic interests in the region. All of the empires and powers possessed these economic concerns, and as such preserving the Chinese empire under the Qing, regardless of how ineffective or obsolete it was, was ideal in the eyes of investors as well as the statesman.

To conclude then, the various powers in China could not and would not colonise it for their own unique reasons. However, they all shared a common fear of exclusion at the possibility of "dismembering" this considerably bit of territory. So if we go back to OP's suggestion in their explanation of the question:

And it seems like the bragging rights of ‘I colonized China’ would be right up the Europeans alley.

However proud the Europeans were of their imperial possessions, such bragging rights would be hollow remarks. A nation would only be able to make that remark of "I colonized China" after years of bloody warfare with both local and foreign forces, economic strains back home and within China itself, and face the unpleasant prospect of having to subjugate and administrate over a vast territory formerly ruled by a fairly advanced power-holder. Gilbert Reid on why the partition of China seemed the worst possible outcome even back in 1900:

"Each nation, while anxious for more influence, is opposed to the increased influence of any other nation. The whole territory of China presents so many opportunities for foreign enterprise (and influence) that all prefer competition to exclusiveness and dismemberment."

Hope this response sheds some light on the matter, and feel free to ask any follow-ups on this interesting geopolitical problem of the past as you see fit.