They tried, but it was never a military possibility. The US, France, and, to a far greater extent, Russia, attempted to impose their will on Korea but failed due to either Korean resolve, the threat of Qing or Japanese intervention, or both.

Western imperialism in Asia had not one, but two peaks. In 1860, the Qing military, distracted by civil war, underfunded, mired in corruption, and over-bureaucratized utterly failed to stop the British and French from occupying Beijing and burning the Summer Palace. This disaster led a clique of reformers led by Empress Dowager Ci Xi, Prince Gong, and Zeng Guofan to seize power in a coup. For the next 25 years, the "Tongzhi Restoration" clique defeated countless rebellions, reconquered Xinjiang, and built an army that defeated the Russians in 1878 and the French in 1885. The Qing military resurgence in this period was for a long time underappreciated by historians, but was fully appreciated by contemporaries who perceived the country to be a rising power. As a result, there were effectively no Western colonial accomplishments in Northeast Asia for the entirety of this era - the aggressive gunboat diplomacy of the 1850s and early 1860s was replaced by limited expeditions, moderated by the threat of Qing intervention.

The two such expeditions to Korea - the French and American Ganghwa expeditions - by diplomatic necessity committed a very small force for a short amount of time. While both militarily succeeded in conquering Ganghwa island off the coast of Seoul, neither deployed enough forces to capture the capital itself. The Korean court - far more conservative (and resolute) than that of Japan or China - obstinately refused to offer concessions to either party, and both countries were forced to back down. It is critical to note when evaluating the small scale of these expeditions that Korea, unlike Southeast Asia, was being subject at the time to Qing imperialism. The Qing court was in the process of revising Korea's loose tributary status to that of a Western-style colonial dependency, acquiring commercial concessions and intermittently deploying troops. Any larger or more ambitious adventure would certainly have led to a wider conflict, as the French expedition to Vietnam triggered the Sino-French War.

Western power in Asia would witness a "second peak" following the Qing's humiliation in the First-Sino Japanese War. Until recently, scholarship surrounding the decline of Chinese forces between 1885 and 1895 has been limited, but research over the past few years has allowed a more coherent picture. Following the Empress Dowager's retirement in 1889, Guangxu Emperor, influenced by his chief advisor, Grand Tutor Weng, slashed military funding considerably. The rationale for the cuts were twofold: first, Guangxu's clique did not share the Empress Dowager's fear of Japan. Second, the army and navy were used by Ci Xi and her allies as conduits to embezzle funds. The result was a total halt to naval buildup and an underpaid army which, in the words of a contemporary journalist, "had no inducement to stay and be killed". Equally important was Japan's excellent espionage work: Tokyo's spies had procured an asset in the Qing telegraph office, Chang Yinhuan, who referred to them all Qing troop movements.

Far from lionizing Japan, the outcome of the Sino-Japanese War convinced European powers that Asian countries in general were harmless. The much feared China threat before 1894 rested on the assumption that even a marginally competent China, due to its immense manpower, could prevail over Western powers in a long war. Japan, a country with a smaller population than every single Western Great Power, was no threat even if it achieved equal military competence to Westerners. At the time, the idea that Japanese soldiers and officers would exceed the capabilities of the Russians a decade later was inconceivable: the Japanese army, Western-advised until 1888, was widely perceived to be a passable but still inferior clone of Europe's forces.

After the war, European powers re-asserted themselves in Northeast Asia, briefly leading to the possibility of Korea becoming a protectorate of Russia. Britain, France, Germany, and Russia forced the Qing to grant them new concessions, perceiving a general power vacuum in the region. Russia, which stayed out of Korea out of fear of Qing in the decade and a half following their defeat at Ili, perceived no such threat from Japan, and turned the Korean court into an effective client state. The small Korean army fell under Russian command, and Russian commercial interests were dominant in the northern half of the country. For his part, Korean King-cum-Emperor Gojong, having lost Chinese protection, was all too willing to assent to counterbalance Japan.

Japan's upset victory in the Russo-Japanese War put an end to this short-lived Russian influence project and placed the Korean peninsula firmly under Japanese control. In 1905, the Eulsa Treaty made the country a Japanese protectorate, while the Taft-Katsura Agreement procured American support for Japanese occupation, after which point any further colonial penetration by other powers became impossible.

Sources:

Story, Douglas. Tomorrow in the East.

Huang Zhangjian. A Study of the History of the 1898 Reforms.

Ivanov, Alexei. The Russo-Japanese War 1904-05.

Duus, Peter. The Abacus and the Sword: The Japanese Penetration of Korea

Qi Qishang. A History of the 1894 Sino-Japanese War.

The Canadian Spectre, Volume 1, 1878 <The brief Russo-Chinese War over Ili in 1878 is almost entirely unknown in the West. This is the only English language source I could find that talks about it, besides the source below which devotes 1 paragraph.>

Jowett, Philip. China's Wars: Rousing the Dragon 1894-1949.

Driscoll, Mark. Absolute Erotic, Absolute Grotesque: The Living, Dead, and Undead in Japanese Imperialism.

You can't really separate Japanese/Chinese/Korean history, especially during this period. So this might read more like a Japanese history response, but I promise you I'll get to the heart of your question.

Japan's emergence as a colonial power following the Meiji Restoration was partly in response to Western powers imperialism. Japan organised a national bureaucratic government and a national economy following the Meiji Restoration and from the earliest stages the "Korea question" was at the forefront of Japanese officials' minds. Keep in mind that the only attempted invasion of the Japanese homeland was by the Mongols and was launched from the Korean peninsula. The Japanese were well aware of this and of the strategic importance of Korea.

Kido Takayoshi- considered one of the three founders of modern Japan- wrote that an assertion of policy towards Korea "would instantly change Japan's outmoded customs, set its objectives abroad, promote its industry and technology, and eliminate jealousy and recrimination among its people." This was written in 1869 when the new Meiji government wasn't as powerful and conslidated as it would be just 15 years later, so scoring a diplomatic win and instilling some trust in the new government would have been crucial. What to do with Korea become a major issue within the government, and much could be said about the topic of Seikanron, which was the proposed military invasion of Korea that was ultimately shot down in the 1870s. There's no need to delve into that here, but by the 1880s and 1890s Western imperialism spread rapidly with Korea, China, and Japan being some of the only Asian countries not colonised by Western powers as you rightfully observe.

One point to keep in mind here was that Japan was actively trying to become a colonial power at this point, and what set Japan apart from colonised regions by the 1890s was their strong centralised bureaucratic government. The last serious challenge to their authority came in the 1870s during the Satsuma Rebellion (which inspired the film The Last Samurai, as an aside.) In the 1880s Japan was purchasing up to 90% of all Korean exports and exporting an enormous amount of Western goods to Korea. It was clear by this point that Japanese government officials and the populace began to see Korea as part of their sphere of influence. They were also aware of Russian imperialist expanisionism increasingly making its way further east.

In 1894/1895 the first Sino-Japanese war occurred, and without going deep into this, Japan's resulting victory and acquisition of Taiwan added to the emergence of Japan as an actual imperial power. It went a long way in stifling dissent, bolstering expansionist rhetoric, and justifying militarism and increased defense spending at home. The resulting "independence" for Korea was seen as an opportunity by Japan to tighten its grip on the peninsula, and it did just that. As historian Mark Peattie writes in The Japanese Colonial Empire, 1895-1945, "the aggressive movements of Japanese forces into Korea, China, and Micronesia was as much due to the absence of effective power to resist it as it was to specific policies and planning."

We could go deeper after Japan officially annexed Korea in 1910 and how it was exploited primarily for food for Japan and what that relationship looked like- especially Korean resistance- but in sum, European powers never colonised Korea because Japan never gave them that chance. Russia had imperial ambitions for Korea, but that was effectively ended with the Russo-Japanese war in 1905.

​

R. Myers, M. Peattie (eds), The Japanese Colonial Empire 1894-1945 (Princeton, 1984).

lriye, "Japan's Drive to Great Power Status", in The Cambridge History of Japan, vol. 5.

M. Mayo, "Attitudes Towards Asia and the Beginnings of Japanese Empire", in G. Goodman (ed), Imperial Japan and Asia (NY, 1967)

Young-Soo Chung & E. Tipton, "Problem of Assimilation: The Koreans", in E.Tipton (ed), Society and the State in Interwar Japan (London, 1997).

As a follow up, couldn’t the argument be made that Korea was colonized by western powers starting in 1945, and lasting in the North until Soviet power was displaced by Chinese power (say by 1960), and in the South probably until about 1980?

Edit: I’m phrasing this as a question instead of a statement - this is my purely anecdotal impression, and I would be curious as to what the post-War academic sources say.