If you're a non-citizen of Latvia, and acquire a citizenship of another country, do you automatically lose your (right?) to be a non-citizen of Latvia?
Yes you lose the non-citizen status. See https://en.m.wikipedia.org/wiki/Non-citizens_(Latvia)
Specifically the 3rd prong of the following: The non-citizens are "citizens of the former USSR (..) who reside in the Republic of Latvia as well as who are in temporary absence and their children who simultaneously comply with the following conditions: 1) on 1 July 1992 they were registered in the territory of Latvia regardless of the status of the living space indicated in the registration of residence, or up to 1 July 1992 their last registered place of residence was in the Republic of Latvia, or it has been determined by a court judgment that they have resided in the territory of Latvia for 10 consecutive years until the referred to date; 2) they are not citizens of Latvia; and 3) they are not and have not been citizens of another state." as well as "children of [the aforementioned] if both of their parents were non-citizens at the time of the birth of the children or one of the parents is a non-citizen, but the other is a stateless person or is unknown, or in accordance with mutual agreement of the parents, if one of the parents is a non-citizen, but the other – a citizen of other country"
If you acquire citizenship you become a citizen of another country. This causes you to lose this status of non-citizen.
Why not just become naturalized in Latvia and then become a citizen of another country?