This is not quite right, legally speaking. Currently Youtube and other platforms enjoy the protection of Communication Decency Act Section 230 [1]. In particular, the passage "No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider".
This passage has two side effects. One is, those platforms enjoy a liability protection - they cannot be sued for the content published on them, as they are not 'speaker/publisher' of the content (the original poster is). The other is, those platforms should not enjoy the freedom of speech protection - as they are not, legally speaking, the 'speaker/publisher' of the content.
The 'freedom of speech of Youtube' argument runs counter to Youtube's best interest - immunity from civil liability, as granted through the Section 230.
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