I do feel maneuvers fill this niche. Using the cover rules is a neat idea.
The maneuvers dip the fighter's toes in the water. I'd push the fighter into the pool.
I would not restrict a fighter's battlefield abilities to limited uses per day. There is not magic that they are depleting. This isn't a spell. This is them being skilled at fighting, using the terrain, knowing how to anticipate enemy actions, and knowing their allies. Why should that be limited?
I'd allow them abilities they learn, and once they learn them they can use them an unlimited number of times. They'd learn a certain number of abilities at first level and gain access to better ones as they advance in level. You'd get 2 plus your intelligence modifer abilities at first level and more as you advanced - with certain abilities requiring a minimum fighter level to acquire them. Off the top of my head here are some ideas I'd toss out there (Note: They are not balanced against each other and som would require higher levels).
Shield Other: If a ranged attack targeting an ally passes through a space you occupy, your ally may add your shield bonus to their AC.
Switch: If you enter a space occupied by an ally, your ally may enter the space you vacated.
Shoulder the Burden: You may grapple an ally as a free action. When grappling an ally you may move your full speed and pull them with you.
Careful Carnage: You may use a bonus action to create a 15 foot by 15 area of difficult terrain by breaking an object you carry or an object within reach. The difficult terrain area must be in or adjacent to the space occupied by the object.
Dynamic Leaper: When you jump at least 10 feet your speed is increased by 10 feet until the end of the round.
Athletic Jumper: Treat your strength score as if it were 8 higher when calculating the height and length of your jumping.
Fast: Your speed is increased 5 feet.
Outmaneuver: When an ally is on the opposite side of an enemy from you, that enemy cannot use an action.
Find Weakness: You may mark a target you can see as a bonus action. Any ally that can hear you gets a +1 to attack rolls against that target while you can see the target. Your mark ends when the enemy dies, or after 1 minute.
Master of the Bull: When you execute a shove as an attack you may make an unarmed attack on the target as part of the attack.
Big Gulp: You may drink a potion yourself or give one to an ally as a bonus action.
Bloody Business: If deal damage to an enemy that enemy has disadvantage on stealth checks for 1 hour or until healed.
Rousing Cheer: You may use a bonus action to inspire your allies. They gain advantage on saving throws against abilities that would frighten them for the next minute.
Hodor: You may create barriers out of objects you carry or within reach as a bonus action. The barrier requires an athletics check with a DC of 8 Plus your proficiency bonus plus your intelligence modifier to move. The barrier makes a 5 foot by 5 foot area unpassable and can block doors from opening into them.
Taunt: As a bonus action you may taunt an enemy that understands you. That creature must make a wisdom saving throw with a DC equal to to 8 plus your proficiency bonus plus your intelligence modifier. If they fail, the creature is distracted and has disadvantage on perception checks for the next minute unless those checks involve you. They have advantage on perception checks against you.
Perfect throw: As a bonus action you may fling a tiny object up to 30 feet in order to attempt to activate a switch, knock over another object, position the thrown object safely (such as tossing a potion on a table across the room without breaking it), or do anything else the DM allows. When you make the attempt, make a sleight of hand check. The DM will set the DC, but it will generally be between 10 and 20 with 20 being the DC for a seemingly impossible task.
Study Scrolls: You have learned just enough about magic to use a scroll. You may use cantrip and first level scrolls as if you were a member of a spellcasting class of your choice.
Bandage Wounds: You may bandage the wounds of an ally if you have a healing kit and spend 1 minute. When you do so, the next time the ally regains hit points, the ally gains healing advantage for 1 minute. While they have healing advantage they roll twice for hps regained and use the better of the two rolls.
Direct the Path: When you have unusued movement at the end of your turn you may gift it to an ally. That ally must use it before the end of their next turn.
Coordinated Confusion: You may mark an enemy that you can see as a bonus action. Allies that can hear you gain +1 to AC and saves when that enemy attacks the ally or forces the ally to make a saving throw. Your mark ends when the enemy dies, or after 1 minute.
Threat Assessment: You may use a bonus action to learn the resistances and vulnerabilities of an adjacent enemy.
Assess and Capitalize: You may use an action to force an enemy to make an Intelligence saving throw with a DC equal to 8 plus your Intelligence Modifier plus your proficiency bonus. If they fail, that opponent must select how they will use their action on their next turn now, including any targets that need to be specified. When they take their turn they may use their action only to perform that action, to dodge, or to dash.
Battle Dancer: You gain one legendary action you may use each round. You may use a legendary action to move 5 feet without provoking an opportunity attack, or to move an adjacent ally 5 feet without provoking an opportunity attack, or to allow a prone creature to stand.
Steady Foot: You and allies that have been adjacent to you during the turn have advantage on acrobatics checks to keep their balance until the end of the turn.
Find the Hole: You and allies within 15 feet may elect to Find the Hole. A create that has elected to Find the Hole may move into a space occupied by an enemy, but must vacate the space(s) occupied by the enemy before they stop their movement or they are placed prone in the nearest empty space. The normal rules for opportunity attacks apply.
Wrestling Team: You and allies within 15 feet may use an attack to attempt to escape a grapple. Additionally, you and allies within 15 feet may use a bonus action to attempt to restrain a grappled enemy. The grapples foe may make an atheltics or acrobatics check to avoid being restrained with a DC equal to 3 plus your strength modifier plus your proficiency bonus.
Plot the Path: As a bonus action you may designate a number of spaces equal to your intelligence modifier. You and allies that can hear you cease to treat those spaces as difficult terrain.