Yes, people can show their displeasure by not supporting the team. It is a business.
1. But in Cleveland's case, it's pointless. And that's why they're changing it and Atlanta isn't changing theirs.
Their logo was blatantly offensive as a racist caricature they HAD TO CHANGE. The Block C was generic and lame. They HAD TO CHANGE their brand/image in some capacity with something to use after retiring Chief Wahoo.
If you HAVE TO change the image/brand to not use racist imagery, just finish the job of eliminating that negative aspect of your brand.
They could have made a non-offensive logo of a C wearing an authentic looking headdress, eliminating the racist caricature; but you don't get any bang for your buck out of that:
You have people who are going to hate change because it's a change from tradition (just like they have now)
You have people who are going to say the change was positive, but not enough on the "offensive nickname" front.
You can't avoid having the first group no matter what you do, but a total rebrand eliminates the second group. So just do a total rebrand and in the long run, you'll be fine.
The old timers who hate the new stuff will gradually diminish over time as they (a) get used to it, or (b) die off and are replaced by younger humans on the planet who don't know the difference. (Much like how Soccer has been accepted in the US when it was actively hated by a large population in the 90s).
2. The premise of your argument is that "People can show their displeasure by not supporting the team because it's a business?" The team made the business decision to not alienate potential customers with an offensive name. That's the right business decision.