TF2 doesn't have any BR modes. KOTH is a standard arena shooter mode, with teams and respawning; "KOTH" only signifies that the objective is to control a static area in the middle of the map.
The closest that TF2 gets to a BR is its "deathmatch" or "elimination" mode, I don't remember the name, in which there is no respawning and the last person standing wins. However, this is very different from a BR; the maps are still tiny, there is no loot to upgrade your character, and there is no "circle", which is a wall of damage that shrinks to force the remaining players to move towards each other rather than staying put.
The closest that TF2 gets to a BR is its "deathmatch" or "elimination" mode, I don't remember the name, in which there is no respawning and the last person standing wins. However, this is very different from a BR; the maps are still tiny, there is no loot to upgrade your character, and there is no "circle", which is a wall of damage that shrinks to force the remaining players to move towards each other rather than staying put.