I'm so on board with renaming the franchise the "Spiders" that I haven't even given it a second thought. I don't care that the 1899 Spiders were the worst in professional baseball history, and all the accompanying hexes that would invariably come with that, the name is cool as f**k and outside of the University of Richmond, I can't think of a single sports team - professional or collegiate - that uses the "Spiders" as its moniker, despite how badass a spider is.
Chiefs and Braves aren't big issues but the imagery and "legacy" behind it are. The tomahawk chop, the chants, the previous team logos. I wouldn't blame them if they want to wipe the slate clean
The Atlanta Braves seem to be going out of their way to erase any Native American imagery from their franchise, and if push came to shove, they could remove the tomahawk from their uniforms and continue to be called the "Braves" but have the moniker be a show of respect to the armed forces, or police or fire or EMS.
As for the Chiefs, the team nickname could stick but the issue seems to be the "tomahawk chop" chant and the pre-game beating of the drum, as both could easily be interpreted as cultural appropriation. Other than those in-game routines, I see no reason why the "Chiefs" nickname is a problem.