I may be biased because I’m so addicted to reading that I will read Marmelade labels at breakfast if I don’t have anything else to read I’ll be in my office shirt. I often ride/drive around and some word pops into my head that I read unconsciously of the side of a truck or a sign by the road. Reading will build an unconscious vocabulary that you’ll need to compose prose. I can immediately detect when authors have been abusing their thesaurus to make up for their lack of vocabulary because they miss the nuances between the synonyms when they replace their simple prose with more exotic words to sound more intellectual. And that is just words and their usage – reading voraciously as a writer will also teach you about timing and rhythm, suspense, form, and structure. You’ll take that all in through a sort of osmosis, which will help you when you’re composing your own fiction.
Well, there cannot be a binary answer for this but surely reading and writing add a lot to making a better speaker. So the answer to your question can lean towards YES in my opinion. Reading more and more creates in-depth knowledge about a certain topic. It also hones one’s perspectives towards the said topic and also the incidences in relation to the same. Reading develops the vocabulary of a person and makes her/him understand the subtle differences between the words. This makes us choose the right words to convey our thoughts. When one reads more one becomes slowly and steadily well conversant with the verbal communications. It also makes us patient. All these aspects are very well necessary for a good speaker. This will make a politically correct statement from the speaker and she/he can give a better overview on the topic than others.