Yes, it would be a big improvement!

Good writers nowadays rarely use "a man" to refer to "anyone" for the following reason.
1. It is gender biased. That is why we use "a sales person" instead of "a salesman" , "he/she" instead of "he," etc.
2. Because of 1, in modern context, "a man" suggests the meaning of "as a human being," which is an overstatement here as SanGabrielhas commented.