For example, I can be shooting at someone in front of me while seeing shots on the mini map behind me. Get the kill, turn, get the second kill.
Here is the same example from the 2nd squid's perspective: You are using your roller and you see a player shooting your teammate with his back to you. You roll towards him expecting to kill him, but after he killed your teammate he quickly turned around and killed you.
If they added a map on the screen then it would hinder the stealth and sunrise tactics in the game, escpecially the rollers. Rollers won't be able to get kills because the enemy will see their big paint trail on the map. Any roller painting ink on the ground might as well have a sign over their head saying "come get me! I can't fight back from a distance!" I know what you mean by map awareness, but in other shooters you don't have to worry about the enemy knowing where you are unless you shoot, however in Splatoon whenever you put down ink, which is almost all the time, the enemy will know where you are.
Although you have the same advantage, the game will become about killing the other players because you want to stop them from laying down ink. The mindset will go from "ink as much territory as possible and if you find an enemy, kill them," to "head toward the enemy players and kill them, then ink as much territory as possible." The game will become kill first, ink territory second. The only use for the map is to super jump, which helps you get back to inking faster, and locating enemies. The super jump is only used ever so often, so you don't have to see the map constantly, and locating enemies isn't the priority of the game so the developers made it so you don't see the map constantly, because that will encourage killing enemies and not inking territory. There is only one benefit to putting the map on the TV and that is to help you locate enemies and kill them, but that is not the priority of the game. The priority is to ink territory and the developers designed it such that it encourages that mindset and hinders the other.
TL;DR: Putting a map on the screen will encourage killing instead of inking territory, which is not the direction Nintendo wanted to go with this game. It will also hinder most of the stealth and sunrise tactics in the game, escpecially for the rollers.