First I'm going to simplify games to their core fundamentals in order of observed priority from gameplay footage.
CoD Black Ops III: Kills, Coordination, Communication
Overwatch: Objectives, Kills, Teamwork
Battleborn: Kills, Survival, Coordination
Halo 5: Movement, Kills, Rambo (Not fighting as a team)
Gigantic: Positioning, Kills, Teamwork
Paladins will probably be similar to gigantic too early to tell what it's core fundamentals are.
None of these games represent a competitive team atmosphere for me since the average (read:most important) fundamental of most of them is straight up getting kills.
Kills are not king.
Kills have their place in objective based game modes which is solely denying objectives to your opponent.
As soon as you reward kills with points directly added to your teams overall score to win and/or by making denying objectives easier for those who have the most kills the game mode becomes TDM draped in a tablecloth of whatever the game mode's name is with salt and pepper objectives that focus the gameplay.
Here are some examples of competitive action games from my perspective:
Lovely Planet - Objective: Have the fastest time
DeadCore - Objective: Have the fastest time
Rust - Objective: Survive Alone
Payday: The Heist & Payday 2 - Objective: Complete Several Objectives ASAP
What do all of these games have in common?
Objectives are king, they are not flexible at all on that.
When you secure an objective it is set in stone and there is no going back only forward. You don't get points added to your overall score for getting kills. Killing can be entirely unnecessary as well.
Throwing two teams at each other and calling it competitive does not make it competitive. Competition is exclusively from the players perspective on how they are interacting with others or the game itself.
When you compete against others the goal should always be clear:
BE the best - This does not mean be at the top of the leaderboards on a match by match basis
BE the fastest - This does not mean hesitating or focusing even for a moment on anything BUT moving from point to point
BE the last man standing - This does not mean hiding in a corner until the last minute
BE a team - This does not mean getting the most kills, strong communication, or perfect coordination
When you tell a player to occupy a space on a map (capture and hold) or to get a point on a map (capture the flag) then you give them a point for doing so.
If they die they should not lose points nor lose the ability to continue to gain points. They succeeded, reward them. If it's to "defend a position" then every kill they get while occupying that point should give them points or every enemy they kill within that point should give them points.
"Action" is easy to embellish, but Competition is not.
TL;DR - Leaderboards are much more competitive than throwing players at each-other.
