Years ago I worked for a company that had ads appear on Breitbart. Retargeted display, not a direct buy on the site. People would screenshot the ads and light up our Twitter complaining we were directly supporting hate speech etc, so we asked our demand partner to stop buying there to avoid negative engagement on our social.
If your site gets a critical mass of negative attention brands can shut down your ability to be commercially viable, and effectively censor the content in a way.
Yes they do and they should. By showing your ads on a website you are financially supporting that website. If the website allows nazis to fester companies should absolutely care to not associate with said website.
But for example, advertisers avoid youtube videos that use slur words, or other non advertisers friendly topics. Its not like they stop advertising on youtube just because a video that said fuck exists. So who the hell associates the ad before the video to the video itself?
But, as you said, advertisers avoid that sort of content. They're willing to still put the ads on YouTube because they know they can avoid their ads being shown next to certain topics.
If you see
buy Pepsi
Insert racial slur
That's not going to give you good associations with the brand
If the video itself advertises you - yeah, i absolutely see that. If its just one of those ads youtube slaps before the video starts - nope, i cant see it.