Before starting, I should note that I will take Japan as the paradigm case for comparison. This is because 1- I do not know that much about the cases of South Korea and Taiwan and 2- much of their Westernisation occurred during the Cold War with a communist twin (North Korea and Communist China) next to them. I believe these are factors that made things quite different in these cases.

A key to answering your question is actually contained right inside your question itself: 'proximity to the Western world'. The Japanese, to give one example, did not fight against the West for centuries, did not imagine themselves in a sort of endless conflict with them, did not actually find the West supporting others against Japan (until after modernisation, that is).

Before the Republic, some Turks saw themselves as an antithesis to the West. Granted, by the 19th century, this belief was sufficiently weakened to have nearly all intellectuals agree that some sort of modernisation was needed. In the late Ottoman Empire, there were many modernisation attempts such as the Tanzimat period. Yet, none was as radical as the Kemalists and the radical Westernisation we associate with the Japanese or the Kemalists today was indeed a novelty in Turkey in the 1920s, despite the fact that some intellectual circles (few in number) did advocate for radical Westernisation before.

You have, therefore, this feeling against the West, the West representing the Crusaders, and so on. It was not easy to convince a Muslim population to become Westernised in part because of this. While traditionalism was and still is a thing in Japan, I do not think the obsession with the West and an antithetical relationship between Japan and the whole of the West is something that profound and widespread. To this, add that many tenets of radical Westernisation were in opposition to Islam or how Islam is generally practised, at least. Take the legal equality of males and females, take the age of consent, the old Ottoman alphabet (derived from Arabic and Persian) being associated with Islam... All these empowered Islamism time and again, which in turn prove itself to be the most consistent enemy of Kemalism and radical Westernisation. This does not of course mean that like some European thinkers believed in the 19th-century Muslim people are unable to progress, because of a genetic or quasi-genetic inadequacy. This just means that Islam, at least as practised by a significant portion of Turks, had/has tenets in opposition to Westernisation. Islamism in Turkey was deactivated, pacified by Kemalists from the 1920s to 1950s, or 1960s. It was not, however, eradicated. Once more, the issue here was probably lack of means than a belief in ideological diversity, considering Islamists and Kemalists never had much tolerance for one another.

Despite all these, you are right that the will was there. Turkey under Atatürk did intend to become a democracy at some point, but it is difficult to describe Atatürk's rule as a democracy exactly. There were elections but these were heavily regulated by the Party, the main function of which was to prepare Turks for a democratic future to come, probably after signs of Islamism and communism disappeared. However, there were technical obstacles behind this will. Ask yourself this. You are convinced that doing things a certain way will be very good for the people of your city. Let's grant that you are sure and you are actually right about this. What do you do then? You can try to convince them, for which you need to reach them, to have ample information about what kind of people they are, what they like, what they believe, and so on. You need means of communication so that your voice would be heard. Now, you have the social media profiles for the first (or Cambridge Analytica if you really mean business) and various means of communication for the second today. How about then? The Kemalist network of intelligence was necessarily limited by the human element. It necessarily depended on local elements to provide information about locals, besides the visits of higher figures such as Atatürk or İsmet İnönü to towns and villages. While it is unthinkable that the Kemalists leaders thought people in far-away towns or villages rejoiced with the fruits of the Kemalist revolution, they simply could not know instantly the beliefs operating there, and the degree to which radical Westernisation was accepted. Some people did warn them that especially in villages the situation was as if the Kemalist revolution did not happen at all (Yakup Kadri Karaosmanoğlu is an example), but the fact that the majority was not that pessimistic and tended to believe the Islamist 'subversives' to be a rather loud minority did not help. When it comes to reaching these people, they needed money. Kemalists did open a whole lot of new schools, even in villages. However, this was not enough. Kemalists did not have an all-encompassing propaganda machine like the one Nazis, Soviets, or even the Italian fascists had. In the 1930s, there was some interest in the propaganda methods of these countries, especially of Italy, but again, they were scarcely implemented as they were. While education and especially healthcare activities were significant, especially in comparison to the preceding periods, they never reached levels found in Europe.

There is also the issue of duration. The Meiji Restoration happened between 1868-1912, whereas the period of High Kemalism is at best between 1923-1946. In 1946, Turkey started to transform itself into a multiparty democracy, which, as any revolutionary would tell you, is more a system designed to represent what is than to move towards a goal contrary to what is. That is, a liberal democracy is not particularly suitable for conducting revolutions, a point taken by different people to be a merit or a shortcoming of it. In democracies, often, political parties must embrace (or pretend to have embraced) the values and beliefs of the people. Otherwise, they do not have a strong chance of winning the elections. This is what happened in Turkey too, as the advent of the multi-party politics led both to liberalisation and an increased attempt to meet the needs and desires of those sectors of the public that were in the past deemed as subject to enlightenment. For example, in 1949, the religious imam-hatip schools were opened (under İnönü, though the following governments went further in this regard) and a generally more relaxed and liberal attitude towards non-Kemalists was embraced. Turkey did eventually become a democracy, within certain standards, of course, but quite advanced when compared to anything that preceded it or anything around it at the time. Yet, the Kemalist revolution stalled. Leaders like Süleyman Demirel can hardly be considered as Islamists, but under democracy, all is fair game and despite being Kemalists inside, they played with the Islamic feelings of the population, which itself began further augmented in the 1970s as a result of certain demographic and intellectual developments. The result was the rise of Islamism in Turkey.

So, to roll back the film a little bit, Kemalists did not have the same amount of time the Japanese had. In this limited amount of time, they faced a formidable enemy of Westernisation in the form of Islamism. By the 1930s, it was probably very difficult for Turks to believe that Islamism would make a come back; even Islamists themselves were pretty gloomy about the future back then. Yet, the shift to democracy in 1946 and the stalling of the Kemalist revolution that followed turned the tables. It was not as if Kemalists were a 1% elite ruling a 99% hardcore Islamist population. However, considerable portions of Turkey were and are receptive to Islamism of all sorts. The success of the Kemalist Westernisation was the creation of newer generations of Turks, especially in cities and large towns, that were more Westernised (and at the same time more 'Turkish' from a nationalist point of view). There is obviously no hard data on this but we can safely bet that Kemalism led to Turkey having more secular people than ever before. Yet, they did not ultimately win against Islamism. Islamism was an enemy that had deep roots in villages, especially in the Eastern parts of Turkey, where Kemalists found it harder to penetrate. Would things be different had Turkey not transformed into a democracy in 1946 and 1950? That is more counter-factual theory and politics than history. What is certain that radical movements of this sort always take time to train newer generations and brute force, which in turn requires a willingness to use it. I am thinking of the fate of the samurai under Japanese Westernisation here. Add to this the fact that the samurai were a limited elite class of warriors whereas in the Turkish case you need to revolutionise not just, say, imams, but the villagers and the town-dwellers as well. Force and willingness the Kemalists had for a while, but increasingly less from 1946 onwards. Time, they did not. It was not as though the shift to democracy was imposed upon them by Islamists, by the people, or by foreign powers though. It was a government under İnönü himself that left the power to the Democratic Party in 1950, the founders of whom split from the Kemalists' Republican People's Party for reasons of economic policy.