Fact checking itself is not black and white, it is subjective
In many circumstances, individuals are not objectively wrong or right, so the act of removing content or tagging it as inadmissible could reveal the media site as biased.
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Fact-checking and tagging media is not an objective act. From the perspective of a sharer, the content could be accurate, but each person viewing the content could carry a different opinion resulting in their impression of it being extremely different. For example, in recent days, a trump official shared a clip of Biden speaking at a rally, which was subsequently tagged by twitter as "Manipulated Media."  In this clip, Biden was shown stating, "we cannot get re-elect, we cannot win this re-election," which are, in fact, words he actually uttered during his Presidential rally that was later incorporated into Trump's political campaign. Regardless of these words being the exact words taken from Biden's speech, Twitter has declared it as being manipulated in their opinion, presumably due to the rest of the speech being omitted. As we can see, in this context, manipulated is a very broad and graded term since it can encompass any media that is "synthetic or manipulated media that are likely to cause harm." In one sense, the Trump administration was simply retweeting a highlight of a campaign speech of Biden's in a pure form, but in twitter's opinion, this direct evidence is considered completely synthetic and was subsequently tagged. Twitter also has a policy where "we may label and reduce the visibility of Tweets containing false or misleading information about civic processes in order to provide additional context."  Twitter even has an advertising policy stating that "Advertising should not be used to drive political, judicial, legislative, or regulatory outcome" which has come under scrutiny of both Democrats and Republicans alike! These broad policies Twitter employs are inherently subjective and point to the company's inability to accept the fact.
The specific example stated in the Argument section, where the Trump administration shared Manipulated Media of a Biden Speech, can be interpreted differently in various contexts. In the context of Biden's speech in real-time, it was his talking point to bring his supporters together, but in the Trump tweet, they used it to show that Biden had endorsed Trump (where in reality, that was not the intention). In this context, the segment and tweet were, in fact, manipulated.
Rejecting the premises