What you refer to as "necessary" comes from Modal Logic, which is interpreted by many as "true for all possible worlds". In formal Type Theory, this correspond to something provable from an empty context. However, ordinary Propositional Logic has "true" and "false".

You are correct in the way that Propositional Logic requires an interpretation, which usually covers the entire language when given. This means that there is not a unique way to interpret Propositional Logic, hence no unique way of interpreting "true" and "false". However, you can also not exclude the possibility of interpreting these values as literally true and false respectively.

I am not familiar with Benjamin Libet's work, so thanks for mentioning him.

What is language? That is an interesting question. I do not know the answer. However, I know that many people underestimate e.g. Propositional Logic because their brains can't comprehend what an exponential semantics is like. You have to learn it yourself to understand (in my opinion).

I don't think there is anything special about natural language, or any special property, which can not be used as an interpretation of some formal language. Now, the problem might be that what you consider some kind of "intrinsic quality" of natural language is hard to make precise, since you only have natural language to appeal to (I guess?). What do you think?

I wrote about this idea, of the false dichotomy between subjectivity and objectivity, as a topic in formal logic Avatar Modal Subjectivity. The basic principle is that one can talk about "uniform subjectivity" as when a proposition holds necessarily, but not mentioning whether this is "true" or "false". Hence, there are languages that can't talk about objectivity in that sense, e.g. music, where the digital signal 0000... and 1111... in the raw audio channel both means silence.