I do think it says something that every Jew I've met who regularly goes to synagogue believes that a Jew who converts in is fully, ethnically Jewish, regardless of genetics, and can call themself Ashkenazi/Sephardi/etc., and all the Jews I know that disagree were raised with only a glimmer of cultural connection to the Jewish identity and if they're reconnecting it hasn't been for very long.
Ethnicity is decided by ancestry, language, food, culture, traditions, history, social treatment, and religion. Missing one (not speaking Ladino, being atheist, etc.) does not preclude you from an ethnicity. Genetics are not the same thing as ancestry, especially in a society built around tribal affiliations. It is not synonymous with the concept of race.
Converts, gerim, Jews-by-choice, new Jews, you're wholly Jewish. An antisemite does not ask how you were raised before calling you a slur. Nobody decides if you are Jewish enough besides your beit din. You are welcome, you are one of us. Indeed, halacha forbids us from discriminating against you or reminding you of your time before your first visit to the mikvah. It is not inappropriate to remind people of this, if they break those rules.
Jews who are reconnecting, converts do not threaten you. You might actually share a lot of experiences, which is a good thing. A lot of the resources that are utilized in the conversion process are available and useful for you as well, if you want. You both deserve grace for this process.
Jewish values tell us to embrace each other. Do not ostracize one another for doing Jewish things.
