Speaking as a liberal who refuses to identify as progressive, this kind of thing - the idea that e.g. freedom of speech is a legal rather than a natural right, and that the only natural right a person have is basically "to be happy" - is one of the reasons why.
To address your specific point, by itself, speech cannot harm the worth and dignity of any human being. Worse yet, what constitutes "worth" and "dignity" is so extremely subjective and culturally relative, that any such analysis would be the same. Because of that, I dare say that it is a poor basis for legal constructs. We should stick to laws that reflect measurable, objective harm, and use other mechanisms to deal with things like these.
For example, hate speech is best dealt with not by fines and prison terms (which tends to create martyrs out of those people, and boost their propaganda - "if they're trying to silence them, they must be saying something important"), but by voluntary boycotts, public shaming and other forms of social ostracism. This makes the definition of "hate speech" fluid, subject only to the social mores of the given era - as it should be. It also means that no idea can be completely silenced solely on account of being offensive, which is important to ensure true political freedom.