I believe that to be successful at promoting a customer-first culture in your customer-facing teams then you would benefit from teaching your team members servant leadership.
Servant Leadership is one of the terms used often when talking about Agile Project Management and Scrum teams. As often happens with words or terms over used in job descriptions and resumes alike they do tend to sometimes become platitudes and lose their meaning. But that’s another post…. Still I do think it is a skill, as opposed to an attitude or philosophy, and as a result can be taught and developed in people.
The term is often used to describe how people lead internal teams. Leadership that inspires, enables and encourages success. That focuses on removing obstacles. That promotes collaboration. Providing people with what need to be successful and achieve their best outcomes.
Sounds like successful customer engagement to me.