I don't like or use Twitter, but this isn't a fair criticism against them. If Twitter receives a DMCA takedown, then they have to remove the offending content or else they're sticking their neck out legally. If they say, "this is fair use," then they're liable for copyright infringement if they're wrong. You can't expect a company like Twitter to take risks like that if there's any possibility of them being wrong; nor can you expect them to dedicate resources to having lawyers carefully analyze each case. This is just the natural result of the way the DMCA law works: service providers always take things down after receiving a DMCA takedown unless the takedown is so obviously invalid that any court in the world would immediately throw it out.
Under the DMCA, Trump could in this case theoretically send Twitter a counter-takedown in which he asserts that the DMCA takedown was illegitimate (eg. due to fair use), and then Twitter could restore it without liability. I don't know if Twitter actually has a process for handling counter-takedowns, though. (They should.)
IANAL, but it's not clear to me that it is fair use, mainly since Trump isn't commenting on the work itself (ie. the news piece), but rather the subject of the work. If he was criticizing the handling of the reporting, then that'd more likely be fair use. Probably a lawyer could convincingly argue in court that it isn't copyright infringement (maybe due to the excerpt being insubstantial moreso than fair use), but again, if there's any ambiguity at all, then service providers like Twitter aren't going to stick their necks out.
If people were using a decentralized, uncensorable version of Twitter (which technologically isn't even that difficult), then this wouldn't be an issue...
I mean just to add onto this -- which I fully agree with - I don't think twitter is just going to change their algorithm to allow for people who are verified to be allowed to share copyrighted material without it being censored.
This isn't just something that happens on twitter by the way, happens on Facebook, Instagram, YouTube, etc. Nothing to do with Trump.
Un-censorable in the sense of community run? I've discussed this a couple time on the section, but I do think that at first small community run social networks can survive like this yes. But you hit a point where people notice they're able to share anything and get away with it -- here comes the kiddie porn and stuff like this, which everyone can agree shouldn't be shared. So now there is a moderator that removes this, but now these people are looking for ad revenue and the only ad partners that are willing to work with them don't like that they have porn on their website.
I think it's a slippery slope of just going back to control.