Isn't “coming out” and “pride” essentially immodest and incompatible with the Orthodox Jewish values of Tzniut? Shabbat violators may be tolerated but not if they advertise their sins publicly?

Orientation is about identity and living authentically, not about personal sexual experiences. Watch this video to learn more about what LGBTQ identity is.
Pride is about cultivating a healthy sense of self-esteem, not about advertising private information or displaying arrogance.
When someone comes out, one cannot assume that they are telling you that they are or plan to be sexually active. On the contrary, it is harmful to sexualize LGBTQ youth by assuming that they are referring to sexual activity instead of identity. The laws of modesty apply to LGBTQ and straight people alike.
Being openly LGBTQ is not at all comparable to publicly violating the Shabbat, because simply “being” gay is neither a sin nor a public “action”. The analogy to Shabbat would only hold for an individual who chooses to have sex in public. Comparing “out” LGBTQ people to Shabbat violators essentially accuses LGBTQ people of public displays of sex and is highly insulting.