I once heard an interesting remark in a YouTube lesson by Rabbi Sherki, which is also Correct in this case. I’ll summarize what he said and expand on it a bit now—don’t worry, this isn’t an interpretation or distortion of his words, just an addition that fits the spirit of the question—based on reality. The process of conversion (excluding IDF conversions, and conversions of Israelis with a very minimally traditional-religious lifestyle at best, without getting into the question of whether these conversions are valid or not) is a difficult and traumatic process. A person undergoes a 180-degree transformation. Think about a newly religious person moving from a secular lifestyle to a religious / Haredi (ultra-Orthodox) one: distancing from friends or often complete separation, changing hobbies, habits, ways of speaking, many times. And think what trauma a righteous convert goes through. You need strong faith to be a righteous convert, because your previous identity dies—your siblings, your friends, your family, your language, your concepts, your parents—you give up everything in one way or another and become a completely different person.
I think that at this point he is getting at the fact that conversion, in its original criteria, is a difficult process. Not everyone, even someone who thinks Judaism is true, is ready for a process this complex. A righteous convert, unlike a newly religious Jew, in my opinion faces many more trials and emotional difficulties.
I think that is why they do not rush to accept converts. They know that a real conversion process is something only a few will withstand, and most will fall by the wayside and return to their old ways.
Judaism is built in such a way that it is a religion that belongs to a particular nation, at least for now. If historically the currents of Germans / Frenchmen / Hungarians of the Mosaic faith—and I am not talking about Reform; in Germany there were Orthodox rabbis who were careful with every minor and major point, but adored Kant, Hegel, and the German emperors to death and metaphorically kissed their feet, and wanted to belong to all of that in the fullest sense of the word—I think conversion would have been an easier process.

Clearly, in the past conversion was a difficult process, and therefore the assumption was that a convert who came was a righteous convert (unless there was evidence that he was doing it for a spouse and the like). The interpretation that they make it difficult for him in order to make sure he will endure may be right, and it may also not be. Simply speaking, they make it difficult in order to make sure he is a genuine convert, and that’s all.
I did not understand the claim at the end. And I also did not understand how all this is connected to our discussion.

Michi, what I meant was this:
Conversion is difficult, in my humble opinion, because it is a complete change of a person’s identity and an erasure of his past. A French person, for example, who wants to undergo a true conversion will be forced to distance himself from his family to the point of only very minimal contact, disconnect from all his friends, from his attachment to his homeland, to his language, to his culture, to his customs, and change patterns of thought and character that are unique to him as a Frenchman. All this stems from the fact that Judaism is a religion with a certain national element, and conversion is not just changing religion as in Christianity and Islam—it is both changing religion and changing national and cultural identity. If Judaism were a religion with no national element at all (and as an example of this I mentioned the Orthodox attempt in Germany and in other places in Europe to create such a model of Judaism), then the conversion process of a patriotic Frenchman, for example, would have been easier, because he would not have had to disconnect from his family, his parents, his surroundings, and his culture. He would remain French and change his religion to Judaism and join a group of people who are indeed of Jewish origin but see themselves as French people obligated to keep the commandments. Simply put, I think that in such a reality the process of conversion would be much easier. Not that taking upon yourself 613 assorted commandments and prohibitions is easy—but in my opinion the hardest element in conversion is the change of identity.
I am not saying that in such a situation millions of righteous converts would suddenly multiply, but I do think that in such a reality it would be easier for a person to go through the mental process of conversion. There also would not be that sense of mutual foreignness between him and the Jewish community he is joining.
True, the interpretation I offered is not at all binding, but in my humble opinion, in the element of pushing away prospective converts, this concern—that they will not meet the demands—really does exist. And also in the literature, one of the interpretations of “converts are as troublesome as a skin ailment” is that they may add sins.

As I said, maybe yes and maybe no. A German or French convert can also easily come to sin, and there is also room for concern that he will fail. Moreover, precisely because Jewish law is not the be-all and end-all, it carries less weight and there is greater concern that he will fail. But one can raise arguments in every direction, and I do not know where this discussion is going.

As an amusing anecdote about the relationship between conversion and willingness for cultural assimilation, I once heard Rabbi Benny Lau say in a lecture that in immigrant families from the former Soviet Union, the children come back from kindergarten with jelly doughnuts and demand that they hold a Passover Seder, etc. And Rabbi Lau said: I asked the grandmother, “So what are you?” She answered me: “I’m a traditional non-Jew.”

Thank you very much for the answer. A question that occurred to me:
“There is an ethnic definition, and that is what determines it.” Is that itself not something that can only be understood as a mere divine decree, or mysticism? Being obligated in the commandments sounds completely detached from the ethnic plane (why do I need to be part of the Jewish people in order to put on tefillin, for example).
What rationale is there for the commandments to have spiritual meaning when a Jew performs them, but not to have meaning if a non-Jew does?

Not at all. There is a certain group upon which the task of observing all 613 commandments was imposed. From considerations of efficiency, it makes sense to choose an ethnic group. There is absolutely no need here to resort to mysticism.