The definition of Tone seems to be entirely based on author's intent. But I remember reading that modern Literary Analysis heavily discourages relying on author's intent when Analyzing a passage. Why then, do we still analyze tone?
When people talk about tone, they're typically not talking about the tone the author intended, but rather, the tone the audience perceives. Tone isn't an absolute monolith intrinsic to a piece of writing - it's wholly a part of the way an individual perceives it. That makes it relevant to a great number of lenses.
To clarify the dictionary definition that you found in comments, there are a couple points worth noting. First, a lot of the definitions a surface search on Google will tell you are colloquial meanings, not literary ones. It's important to be skeptical of a surface-level definition of a complex topic. Especially because, when those definitions are correct, they often rely on words that have different literary and colloquial meanings, and so it's very easy to think a simple definition suffices when in reality it may have just confused you further. (Look up the definition of "text" for a clear example - even Wikipedia arguably gets this wrong.)
Second, in this case, it's actually a reasonable definition - but the writer is a rhetorical writer, not the actual writer. While some tonal information is encoded in what the reader knows about the writer, most of it is encoded in the word choice independent of intent, which is the "writer" the reader perceives.
(It's also important to note, as Hamlet mentions, that trying to discern what the author meant isn't universally a bad thing - it's just not the complete picture, and we need to be very careful not to over-broadly apply that kind of thinking to questions where it's not relevant.)