If the name trivializes or mocks or derides, it should go. If the symbolism is belittling and mocking, it should be punished as soon as possible.
If the name appropriates an ethnicity that subsumes a whole people, it should probably go… unless it's done with so much respect and consultation that those people sign on and give their okay. Which seems highly unlikely.
But if it is instead a profession that provides the name… something that men and women have chosen as their line of work, then the issue, I believe, is whether it is done with respect.
"Gooooooooo Bureaucrats!"
On the level of symbolism, on the other hand, the example images above show that the Redskins have made real efforts to depict nobility while the Cleveland Indians are clueless. Either way, sorry... it's too late. Accept it.
At the opposite extreme is the team name: Atlanta Braves. A "brave" is not strictly-speaking an ethnic group, it could be regarded as a profession. Packers, Steelers, Oilers, Cowboys, Raiders, Pirates… One can argue that the name is a tribute to the same admirable qualities -- agility and boundless courage -- that inspired the creators of the fantastic Crazy Horse sculpture in South Dakota. Efforts have been made to modernize the imagery.
In this case, the issue might revolve around -- is the respect and admiration genuine?
Still, having pointed out that distinction, I do believe that - regarding this controversy - I am in no position to judge or declare right or wrong. If my point about "a profession" were to persuade some Native American people to negotiate an advisory panel with the club, leading to deeply respectful symbolism that satisfied all sides, might the name be saved?
But barring that, well, a name that offends millions ought to go. This isn't "political correctness." It's just courtesy and noticing things our parents didn't.
Same with the Cleveland Indians. For heaven's sake. You Clevelanders already have a new mascot ready to go!
Just call yourselves the Cleveland Grovers!