I'd argue Tumblr and the social justice shift that took place around 2012-13 was widely responsible for this moreso than Twitter. Quite a few actual hobbies were affected by this, but they managed to withstand it better because of the fact they're not fandoms about artistic devices that normal people shun. The furry fandom on the other hand overlapped perfectly with the Tumblr audience. They're rejected by society and are looking for validation for their weird fetishes and poor lifestyle choices. When Tumblr was huge, furries were everywhere on that site. The tagging system, the social media aspects, and Tumblr allowing NSFW content helped keep Tumblr around as an art site as well as it was easy to discover artists.
To add to this some events accelerated the social justice and political shifts. The fall of RMFC and the Alt-Furry trashfire afterwards was a key event, along with Twitter blocklists that were far more effective within the furry fandom. Then there were some high profile incidents of furries being denied art for having the wrong political beliefs or hanging around the wrong crowds at this time.
The end result? Furries are either laying low with their political affiliations hoping that some other furries don't cancel them for it and get them blacklisted from getting art/banned from cons, or they're tripling down on their beliefs. Oh, and you can now find Democratic presidential candidate staffers at furry cons:
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