Its visual equivalent is logo design, if done properly. The mission statement should stand out as well as be a concise capture of purpose. Therefore, undifferentiated phrases about being the best, or putting customers first fail dismally as mission statements.
And whatever the mission statement is, it should work as a marketing tool. It isn’t of any use if it’s only displayed on the wall of the company’s office. It needs to communicate brand purpose to customers and prospects.
The short answer is: yes, a mission statement can capture brand purpose, but you shouldn’t mistake it for brand purpose.