What are some of the reasons for which Catholics are against gender-neutral secular marriage? I get that Catholicism won't acknowledge religious gender-neutral or same-sex marriage (GNM) and why it won't support GNM,

but why exactly does that extend to being outright against (as opposed to having no stand) secular GNM?

Besides possibly the arguments here, what are there?

In case the article is too long for you, here is a response that takes the article apart.
Pope Leo XIII's encyclical on marriage, Arcanum §23, gives a succinct answer to your question:23. Let no one, then, be deceived by the distinction which some civil jurists have so strongly insisted upon—the distinction, namely, by virtue of which they sever the matrimonial contract from the sacrament, with intent to hand over the contract to the power and will of the rulers of the State, while reserving questions concerning the sacrament of the Church. A distinction, or rather severance, of this kind cannot be approved; for certain it is that in Christian marriage the contract is inseparable from the sacrament, and that, for this reason, the contract cannot be true and legitimate without being a sacrament as well. For Christ our Lord added to marriage the dignity of a sacrament; but marriage is the contract itself, whenever that contract is lawfully concluded.His argument is, in a nutshell, that the Church has jurisdiction over marriage, not the State, because marriage is sacred and that is the realm of the Church. In other words: there is no such thing (at least for the baptized) as secular marriage divorced from sacramental marriage. Just because the State benefits from marriage, this does not mean it has lawful authority over it.

Also, to call marriage what is not marriage is contrary to objective truth and the Catholic understanding of human law as flowing from and being in-line with natural law and divine law. "The Canonical Nullity of the Marriage Process as the Search for the Truth" by Raymond Leo Cardinal Burke (the last article of the anti-Kasper Remaining in the Truth of Christ) says:Regarding law, in general, there has developed the notion that the law has no relation to objective truth but is constituted by whatever man, usually the judge, decides. Such a theory was proposed in my homeland, the United States of America, already in 1897 by Justice Oliver Wendell Holmes, Jr. [The Path of Law and the Common Law pp. 1-29]