There is no evidence that studying linguistics has a particularly positive effect on writing and speech, either in L1 or L2. Let us just take the question of effect on L1 writing: and first, you have to be clear about what you mean by "being a good writer". If your goal is to write like James Joyce, J.K. Rowling or Larry Niven, studying linguistics will not help you one iota. Suppose then that you want to write scientific prose, in the field of chemistry or economics. Studying linguistics will not help you, whereas studying chemistry and economics will. But if you plan to write in the area of linguistics, studying linguistics will help.
Merely studying linguistics will not help you much in the domain of writing, but in case your plan of study involves writing, then practice at writing (in linguistics) can aid your writing abilities.
There is some limited potential for knowledge of linguistics (primarily syntax) to contribute to writing skills. An example is the problem of verb agreement, which people might mess up when the subject NP contains another NP and the two NPs differ in number. The normative rule is that the verb agrees with the entire NP, whose number is (usually) transparent on the highest N, leaving aside conjunctions. After a year of syntax, you can probably mechanically figure out whether the NP is singular or plural. Other areas of linguistic study (Indo-European comparative) could help you figure out the normative agreement pattern for bacterium versus bacteria, useful in case you ever have to use bacterium as the subject of a sentence (I don't think I ever have).
I should point out that there is a distinct disadvantage to learning linguistics w.r.t. writing skills: you need to limit your reading to only good writers, which will shorten your career in linguistics. Writing style in certain venues (CLS, BLS, IULC papers) in the 60's and 70's tended to be, ahem, a bit casual.