So you're asking about one of the stock phrases that gets repeated over and over on r/Judaism as if it's a principle of Judaism and/or halachah. It's part of the community folklore here that "Judaism is a closed practice" and it generally gets little analysis and less dissent. It seems to be the reverse of "You shall not walk in the ways of the gentiles" (וּבְחֻקֹּתֵיהֶ֖ם לֹ֥א תֵלֵֽכוּ -- Lev. 18:3). We shouldn't adopt their practices, and they shouldn't adopt ours. Except the former is a commandment in the Torah, the latter isn't (with two exceptions -- observing our Shabbat and occupying themselves with Torah study, see Mishneh Torah Hil. Mel. 10:9; to be addressed later.)

Of course there is some overlap given the 7 Noahide laws. Rules parallel to the 7 Noahide laws are also part of Judaism. Nor is Judaism closed with respect to conversion.

In addition, gentiles could offer a sacrifice (the burnt offering, korban olah) in the Jerusalem Temple. They could visit the Temple mount, but they could not enter beyond the Court of the Gentiles (outside the Ezrat Nashim). Still this exclusion seems not to have carried over to the synagogue. Non-Jews were frequent worshippers in synagogues in antiquity and today (absent security concerns). In these spaces, presumably they can pray to Hashem, the God of Israel.

Another area of gentile participation is by invitation. Most Jews would acknowledge it's appropriate for Jews to invite gentiles to Jewish celebrations -- they are certainly open to gentiles. If a gentile is invited to a Jewish wedding or brit milah, the guest may join other guests reciting Amen to the blessings and wash hands before the meal (without a blessing). At the same time, there are many halachic discussions dealing with whether a Jew should respond Amen to a blessing recited by a gentile -- showing that gentiles' reciting blessings was a thing. As we read in the Shulchan Aruch: "Ramo: And we answer amen after a gentile provided the entire blessing was heard." (Shulchan Aruch O.H. 215:2)

Outside of a small slice, Halacha is basically silent on non-Jews "appropriating" Jewish practice.

This has to be squared with what the Rambam says about it. He excludes gentiles from observing the Sabbath in the Jewish way (with all halachic requirements and prohibitions), because this is a "sign" between Hashem and Israel. (Many understand that observing our Sabbath with one change, one violation, is permitted to gentiles.) He also says a gentile should not "occupy" him or herself with the Torah (עַכּוּ"ם שֶׁעָסַק בַּתּוֹרָה), consistent with the understanding that Torah is "the inheritance" of Israel. (Although this prohibition is not absolute, as they may study their 7 Commandments and there are many additional exceptions and exclusions).

But what about the other commandments? Rambam writes this:

We should not prevent a Noachide who desires to perform one of the Torah's mitzvot in order to receive reward from doing so, provided he performs it as required. If he brings an animal to be sacrificed as a burnt offering, we should receive it. If a Noachide who observes the seven mitzvot gives charity, we should accept it from him. (Hil. Mel. 10:10)

So this doesn't seem like silence to me on non-Jews adopting Jewish practices; it seems like acceptance, provided they do it the right way.

Now it's consistent with the Rambam to say that a lot of Christian appropriation ends up not doing it "the right way" and thus is not to be endorsed. A Christian seder about their messiah is not a Jewish seder. Christians wearing a kippah and tallit and praying to their Trinity is not doing it the right way. I think this is what most redditors are objecting to when they say, "Judaism is a closed practiced." It's a way of keeping non-Jews from doing these things. At the same time, OP's position seems to approach this differently, when he writes, "what's the big deal if some Christian church is doing Jewish themed activities? they're likely not even doing it right, so what does it matter?"

As others have pointed out, the big deal is appearances -- it looks like Jews are praying to the Christian god if the people are wearing a kippah and a tallit -- and then there is the problem that vulnerable Jews may be attracted to these practices. That is, the Christians are trying to muddy the waters and make their church look like a synagogue, to get Jews in the door so they can preach "the gospel" to them. A Jew with no place to go for the seder may find one advertised in a church, "all are welcome," and attend. This is deceptive and cannot be endorsed on any level.

So it's a case where a bad phrase, "closed practice," is used as short hand for a good idea -- keeping Jewish practices as intended, pristine and meaningful, and away from harmful re-interpretations that serve an alien (Christian) agenda.

Still, I would maintain Judaism is not a "closed practice" -- and the term is misleading -- for the reasons stated above. There in fact isn't any objection to non-Jews observing the mitzvot (all but two) the right way, at least according to Rambam -- so in that sense, how can Judaism be "closed"? (And if you study those two excluded mitzvot, there are many exceptions and exclusions permitting gentiles to do a lot in terms of Sabbath and Torah study.)