Answer

This is not a simple question. I had a major argument about it with my study partner a few decades ago. He was very charismatic (a Lithuanian baal teshuva), and he brought masses of people to repentance. Literally dozens of people and families, but sometimes also children without the family. He argued that above age 13 they are adults obligated in the commandments, and we have an obligation to make sure they observe the commandments. And on the face of it, he is right. I think that if you asked rabbis from all camps, most of them would answer that you should not do this, but if you asked why, you would not get an answer. This consideration is ostensibly a simple halakhic consideration to which there is no answer. And nevertheless, in my opinion it is forbidden to do this. I will try to explain briefly.
Before age 13 it seems fairly simple that this is forbidden, because the child is not yet obligated (at most there is an obligation of education, which primarily falls on the parents. Although there is also an obligation on the religious court = the public, when the parents do not do this. But it seems that such a level of obligation does not override the considerations I will bring shortly). But regarding a 13-year-old child, from a halakhic standpoint he is responsible for his actions and obligated in the commandments, and therefore I too have an obligation to bring him to observance of the commandments (“you shall surely rebuke” and “do not stand idly by your neighbor’s blood”).
But as I said, I do not agree, for these reasons:

1. It is not clear that this kind of putting on tefillin really brings them closer to observance of the commandments. At most they are now fulfilling the commandment of tefillin (and even about that I have doubts—because at least those who do not believe in halakhic obligation are not fulfilling a commandment. See my article about causing a secular person to sin).
2. In the culture in which we live, in my opinion it is forbidden to do this, because a child of that age cannot make such decisions, and you are entangling him with his family and parents. One might say about this: better that they remain unwitting sinners.
3. The categorical imperative says that one should not do something that one would not want to become a general law. The question is whether the Chabadniks would accept this kind of secular preaching (or even Lithuanian or Gur preaching) to their children in their schools. Clearly not. After all, they belong to a sect that is not willing to expose its children even to religious people of a different kind. But of course with members of sects there is no possibility of talking, as described there.

By the way, it is well known that Rabbi Shach, who guided adult baalei teshuva, would also tell them not to move ahead too quickly, and would permit them to continue Torah-level prohibitions (and certainly in things that are not a full obligation, like studying in a kollel) so as not to create family conflict. It is hard to base this halakhically, but the fifth part of the Shulchan Arukh says this quite clearly.
Leave the state out of this. It has no standing at all in these matters.