Most superhero names have a silliness to them if examined through too serious or literal a lens. What's a Psylocke? Shatterstar? Boom Boom? Tigra the Were-
Woman etc. Very often it's a vaguely descriptive that sounds "cool". I think Warpath stands head and shoulders above most.
With regard to cultural correction, unlike adjusting Sunspot's name to Beto from 'Berto, Karma's from X'ian Coy to Xuân Cao in the New Mutants, if you take away Thunderbird and don't find something as strong (stronger would be better) to replace it with, you run the risk of divorcing the character from fans visceral connection. Betsy Braddock's fumbling journey to Captain Britain, a mantle she had previously
held, is a cautionary tale in that regard. For many I think Daken will always be Daken, despite the name's translation.
That being said, and in spite of my sig, I can see John Proudstar shirking a superhero name and staying street level and saying "My name is enough". Like his rival X-Man being as much Logan as Wolverine. It fits his saccharine personality, especially in light of his being dead and decidedly anti-establishment. The Thunderbird motif that remains I can see easily explained away as "My brother wore this symbol on his body in tribute to me in death. Now I wear it for him in life." And move forward avoiding the need for a sweeping erasure of history in this case.
(I'm suddenly reminded that Ann early plan considered for Kitty Pryde was that she would take on the name Thunderbird, with it being just a name of former Xaviers student divorcing it from the native connection. I'll have to piece together something about that later...TBC...)