The military does teach servant leadership (or at least the Marines does). The Marine Corps leadership principles:
Know your people: Observe how Marines react to different situations and look out for their welfare
Keep your personnel informed: Provide reasons for tasks and ensure Marines are aware of all relevant information
Set the example: Demonstrate professional competence, courage, and integrity
Ensure tasks are understood, supervised, and accomplished: Give clear, concise orders that are difficult to misunderstand
Explain why tasks must be done: Provide a plan for accomplishing tasks whenever possible
Be alert to rumors: Replace rumors with the truth to stop them from spreading
Build morale: Publicize information about unit successes to build morale and esprit de corps
Make sound and timely decisions: Be able to quickly assess a situation and make a sound decision based on that assessment
Consider advice from subordinates: When possible, consider the advice and suggestions of subordinates before making decisions
Announce decisions in time: Announce decisions early enough to allow subordinates to make necessary plans
I think in general you can be competitive and tactical while centering yourself around small group welfare. It's basically an acknowledgement that you and everyone else are only as good as the group and the leader is most well positioned to affect that welfare on a daily basis.