The Chiefs (similar to the Braves in baseball) have been incredibly fortunate by having a team that has abysmal racial stereotypes in league like Washington that has allowed the Chiefs to escape the lionshare of national scrutiny.
The moment Washington changes their name, that media scrutiny is going to firebomb the Chiefs. It'll be interesting to see how Clark and company respond to it. My gut tells me that the Chop will be a goner, but I'm not sure much else about the team will need to change.
The Chiefs, for their part, have made very legitimate overtures to the Native American community and have a lot of fans hailing from significant Native American communities in Oklahoma. With how the Chiefs actively worked with Native American communities for the drum tradition, it makes me wonder if the Chiefs are/would actively work behind the scenes to perhaps create a new chant (in leiu of the chop) perhaps with creatives from the Native American community. Fans won't like change, but the Chiefs had zero hesitation on pulling their Touchdown song (I can't even remember why) and after a few angry fans spoke out, and got over it, we now have a much better song thanks to Travis Kelce.
I think the Chiefs are forward thinking enough to anticipate the issue here and find common ground, the org has definitely done so before (linked article below):
