Answer

I saw an article that tries to summarize the possibilities for validating such testimony: https://www.tzohar.org.il/wp-content/uploads/38_10.pdf
I have a lot of criticism of what he writes there, but the most problematic point here is that he ignores the difference between testimony that constitutes the matter itself and testimony that clarifies facts about the matter (as with a ketubah). The two are not at all comparable. Most of his sections are not relevant to testimony that constitutes the matter itself. For example, testimony from relatives can be accepted in monetary matters, but not in kiddushin. And by simple reasoning, “they accepted them upon themselves” does not help here (which that article discusses at length), because the witnesses did not come to obligate the groom or the bride. That is true regarding the ketubah, but not regarding kiddushin. Rather, they come to represent the public. So what relevance is there to the groom and bride accepting such witnesses?
I’ll start with the bottom line: a priori, even without sources, it is very puzzling to establish a matter through two witnesses who do not recognize it and are not committed to it. Therefore, in my view, secular Jews do not count toward a minyan, and they should not be given blessings to recite. As far as I’m concerned, they are like adding flowerpots to a synagogue. The same applies to establishing kiddushin on the basis of their testimony. It is like establishing kiddushin on the basis of two flowerpots. This seems to me an obvious logical point, and it is not connected to any source that might be brought. The phenomenon of secularity is new, and usually learning about it from earlier sources, including the familiar categories—child taken captive, unintentional sinner, apostate out of spite, apostate out of appetite, and so on—is mistaken.
Of course, I am speaking here only about atheists. I have an article in Tzohar arguing that atheists are like gentiles. Even if they perform a commandment, they have no commandment, and causing them to stumble is not a prohibition (as distinct from directly feeding or administering something forbidden by hand, which is prohibited). https://www.google.com/url?client=internal-element-cse&cx=f18e4f052adde49eb&q=https://mikyab.net/%25D7%259B%25D7%25AA%25D7%2591%25D7%2599%25D7%259D/%25D7%259E%25D7%2590%25D7%259E%25D7%25A8%25D7%2599%25D7%259D/%25D7%2591%25D7%25A2%25D7%25A0%25D7%2599%25D7%2599%25D7%259F-%25D7%2594%25D7%259B%25D7%25A9%25D7%259C%25D7%25AA-%25D7%2597%25D7%2599%25D7%259C%25D7%2595%25D7%259F-%25D7%2591%25D7%25A2%25D7%2591%25D7%2599%25D7%25A8%25D7%2594/&sa=U&ved=2ahUKEwj-1KuTsPWCAxWvPhAIHYqmANUQFnoECAMQAQ&usg=AOvVaw13Y8Feooy87k16oIjREviJ
They are of course obligated in the commandments, and there is an obligation to try to bring them back to the firmness of Torah, since they are Jews in every respect. But as long as this is their outlook, none of their commandments are worth anything. Therefore putting tefillin on such people in the street is an empty act, just like counting them for a minyan.
By contrast, traditional Jews—and here I define that as people who believe but are lax in practice (there are other kinds of traditional Jews as well)—that is a different matter. Contrary to what people usually think, they are more wicked than atheists, because they know their Master and rebel against Him, whereas atheists simply do not believe and are therefore complete coerced cases, even below the level of a child taken captive. But in terms of connection to the issue of kiddushin, traditional Jews are connected to it and can perhaps constitute the matter, since they believe in the Holy One and in kiddushin and understand its significance. Therefore the degree of wickedness is not necessarily relevant to our question (and one should not make an a fortiori argument here).
Beyond that there is the question of disqualification of a wicked person from testimony, which is what that article focuses on. Clearly, in terms of wickedness in the ordinary sense—not merely someone who commits transgressions, but someone whose personal status is considered wicked—the traditional Jew, by the above definition, is thoroughly wicked. He knows and rebels. By contrast, the atheist is not wicked in any way. He is a transgressor, but as a person he is not wicked, because he is coerced in his belief. From this it follows that if the disqualification of a wicked person refers to wickedness in the sense of the person’s status, then an atheist is certainly not wicked, while a traditional Jew probably is. If the disqualification of a wicked person is because of concern for falsehood, then neither of them—traditional Jew nor atheist—is disqualified, because they are not suspected of lying. That is assuming that testimony which constitutes the matter itself is also disqualified because of concern for falsehood, even though there they are not testifying to anything; someone suspected of lying is personally disqualified and therefore cannot constitute kiddushin either. But if the disqualification of a wicked person is an intrinsic personal disqualification, then at least regarding testimony that constitutes the matter itself, the picture may be the opposite: the traditional Jew is valid and the atheist is not.
I must say in passing that defining the disqualification of a wicked person as concern for falsehood seems very puzzling to me. Why is a wicked person only someone guilty of transgressions punishable by lashes? This is derived from “then the judge shall cause him to lie down and strike him,” but what does that have to do with concern for falsehood? The Sages expound, “Do not set a wicked man as a witness,” and plainly this is an intrinsic personal disqualification. The explanation is probably that we are not willing to grant social standing to wicked people such that matters are established by them—whether kiddushin or legal rulings. In the spirit of: forty lashes on his shoulders, and yet he is fit for testimony?! The fact that testimony honors respectable people means that it is improper to do so for the wicked.
Beyond that, there may also be concern for falsehood in a wicked person, but that is a different discussion. Regarding testimony that constitutes the matter itself, the intrinsic disqualification is important, because concern for falsehood can in principle be bypassed. Simply put, testimony that constitutes the matter sees the witnesses as representatives of the public for this purpose (there is a similar statement in Pnei Yehoshua), and they give public validation to the kiddushin. It is absurd to do this on the basis of secular Jews.
See here in Rabbi Melamed, in the final paragraph: https://ph.yhb.org.il/11-04-05/
Here you have an example that first-order halakhic ruling can also be stringent. We must seek the truth, not leniency and not stringency.

And that is what seems correct to me.