Can critical thinking help you to write better? Or I can rephrase this question, should a person read a critical thinking book instead of a book about writing craft? I personally recommend read both, because critical thinking focuses on a number of aspects related to understanding texts, for example logical construction of a text and biases that can mangle the intended meaning. But books on writing focus on other aspects outside of critical thinking, which are useful for a writer.
You can use other means of learning critical thinking, not the books only. But let’s review how can critical thinking help. As I mentioned logical structure, you can avoid logical errors in your arguments. You can also make diagram of these arguments to check yourself or to use diagrams for something else.
You can also check your text from a reader’s point of view. You can try deciding read or don’t read your texts, when you find them as a reader. If you know your audience you can make a research about them, then adapt your text. So if you expect your audience not to want your text you can look for possible reasons for it. And using this knowledge you can adapt your text and make it better for the intended purpose.
Critical thinking textbooks and writing textbooks agree that each text have a purpose. So if you know the purpose of your text you can edit and improve it accordingly. And if you don’t know you need to find out.
You can also check your text as a critical reader. It can be a friendly analysis to find possible flaws to improve text. Or it can be an analysis to find information to attack your text or yourself. Depending on what responses you expect you can play the role of a friendly reader or an evil reader. Sometimes you can think about possible consequences of your text being published: you can imagine a chain of events, possible forks in this chain, and probability of individual events. If you expect your text to have an impact on society it is a good idea to be prepared for a possible fallout.