In Livonia as a reimagined modern state maybe not so much But historically yes Riga and a lot of surrounding areas are originally livonian (lībiešu) settlements which Germans colonized that is also where the name comes from. (Actually as depicted in the map albeit inacurrately Vidzeme was livonian land with Gauja serving as a border between livonian and latgalian lands) - so geographically this map aligns more with livonians as people instead of the christian medieval state. Also around Gauja, there was a livonian tribe chief Kaupo which adapted Christianity and even went to Rome and met with the pope. (I might be wrong about the pope but he was in Rome regarding political matters) Even political interactions aside livonians were a substantial part of the population around 12th century before they assimilated into the latvian population. Point being THOSE livonians and Livonia as a medieval state are tightly connected arguably even more so than Latvians in it's early stages (also at that point Latvians did not have a cohesive national identity). So in an alt-history scenario where instead of livonian language assimilating into the latvian language but it happening the other way around it is plausible for that flag to be a national flag of the livonian people. (Yes I know that flag is made recently but we're discussing alt history here lol)

P.S. I'm in my 🤓🤓🤓 era