I think you're downplaying the importance of semantics in wildlife conservation massively with this statement. An approachable, easily understandable and relatable name creates a connection between people and birds, and can help bird conservation massively. Giving birds names that laypeople are likely to recognize as birds they have seen can create public awareness of their existence, and thus of their decline and need for conservation efforts. Nobody knows what a "Swainson's Thrush" is, but if hypothetically the name was changed to "Flycatching Thrush" or something else representing Catharus ustulatus's iconic behaviour, more people would realize that they've had these on their property (they're common, but strongly declining) and more people would realize that they're something we need to protect.
A few very real examples of names having an effect on bird conservation:
1) The old name "Oldsquaw" (which is a racist, sexist and ageist slur) for the bird we know now as Long-tailed Duck (Clangula hyemalis), directly negatively affected the communication between conservationists and people in the Alaskan tribes whose lands they were breeding on. Conservation management plans set up by biologists required help and cooperation of these Native Americans, and biologists had to practically beg the NACC (North American Classification and Nomenclature Committee) to be able to change this name because using it in official communication about the project would be detrimental to the plan's success. Long-tailed duck is a globally vulnerable species, that needs all the help it can get.
2) Locally in their breeding range, Kirtland's Warbler (Setophaga kirtlandii) is known as "Jack Pine Bird" or "Jack Pine Warbler", because they depend on young Pinus banksiana to breed. This name immediately makes it clear which habitat and plant the bird is dependent on, and makes it very easy to understand what needs to be done for their conservation. This species was on the brink of extinction 50 years ago, but through habitat management in coordination with locals they are now again self-sustaining, and the use of the vernacular name (and not the official name) proved to be a big part of that. By changing the official common name, an even broader public would be involved in their conservation.
Changing names isn't just an unimportant matter of "academic squabbles". Even if these above examples aren't convincing enough, the notion that this name-changing somehow distracts people from the species' conservation is nothing other than silly: BN4B is not a paid organization, and its core members are doing this project in their free time. If anything, this project (and its backlash) is drawing more attention to birds and bird conservation in general. Nobody participating finds themselves "anti-colonialist progressive heroes", they're just doing what they can to make the world a better place. These effects are real, and mean a lot of things to many people.