I'm not a fan of the "alt-right" or a lot of its subculture. I also don't find it particularly "alt", as much as it is "far" right. It is a lot of the ideas that powered the NSDAP enveloped in the language of the Internet and memes, with the targets broadened from Jewry to non-whites in general. However, I'm also not a fan of suspending these accounts for controversial views if they are not harassing people on the Tw…
Almost all would probably decline. Not just because we wouldn't enjoy the work, but also because it is something that, by definition, would help them and support their goals (according to their judgement – otherwise they wouldn't hire you).
Should people working at twitter not be allowed to make similar decisions? It's not hyperbole – look at reddit, and how it has devolved to something like a lowest common denominator. I'd leave as fast as I could instead of lending support to the toxic community.
Do they have a right that overrides mine? No. Private entities are more or less free to decline any customers, exceptions only where they have monopoly power (telecom, water).
Is fighting them out in the open the more valiant thing to do? No – I don't argue about chemtrails, or religion, or flat earth or evolution. Because it doesn't work. The chance of them convincing me that evolution isn't real is 0. Unfortunately, even though I'm right, my chance to convince them is about the same. Because none of this has anything to do with facts – it's red tribe vs blue tribe, and the right's search for a sense of belonging that Reagan sold to the highest bidder.
Is it going to empower the extremists to ignore them? Considering the ideas aren't new, I can only conclude that the internet has given them increased visibility and therefore helped their rise. If reasonable people decide to curtail their options for propaganda – by hiding them from Facebook, Twitter, Google the way conspiracy theories were ignored by reasonable people in the past, they may just shrink back to their appropriate size of plus or minus 0.