I agree with you, but not fully: if there is no need or budget, then yes, a brand may not be a 1st priority. But in my experience as a designer, an effective brand will be a necessary ingredient to grow, as this is an expectation for a lot of people.
Regarding your brand: You cannot not communicate. What I saw in [1] example it's called a "no-brand" aesthetics: it's very minimalist and very neutral. And to be fair, I like it.