Yes, a Battle Master can give up their single-attack Attack action to use Commander's Strike.
You appear to be confusing the Attack action with an attack.
The Attack action is one of the things you can choose to do with your action; others include the Dash action, the Cast a Spell action, and the Use an Object action.
An attack is anything you do that causes you to roll a die to overcome someone's AC. Trying to hit with a weapon is an attack, but so is using a spell that rolls against AC, such as eldritch blast. You can make attacks without taking the Attack action, e.g. attacking with a spell using the Cast a Spell action, attacking with a magic item using its own unique action, or attack with a weapon using a reaction (e.g. using the Ready action, or making an opportunity attack) or bonus action (e.g. from Two-Weapon Fighting).
The prerequisites for the Battle Master fighter's Commander's Strike maneuver are that you:
- take the Attack action, which always allows at least one attack, and
- forego one of those attacks, even if it is the only one
The rules for grappling and shoving a creature are similar.
They are special nonmagical techniques based on being a scholar of combat. This is explained in the Battle Master entry:
Those who emulate the archetypal Battle Master employ martial techniques passed down through the generation. To a Battle Master, combat is an academic field...
Individual maneuvers explain what you're actually doing:
Commander's Strike: You can forgo one of your attacks... to direct one of your companions to strike.
Goading Attack: You can... attempt to goad the target into attacking you.
In each case, you're mundanely directing your ally or goading (through voice or gesture or just annoying attack placement) your enemy. You're just really good at it because of your battle mastery.
Although I can't find a reference in the PHB, it's generally accepted in D&D that you're doing more with your turn than your single attack. Your PC might constantly be feinting, dodging, making threatening jabs, and so on. Your attack rolls represent those actions that are actual opportunities to do damage. From this perspective, then, a Commander's Strike doesn't mysteriously make your ally move faster; it just gives them an extra opportunity that they otherwise might not have had, just like a retreating enemy lets them make an opportunity attack.
In the games I've played, these maneuvers have been explained however seemed appropriate in the situation and were never assigned advantage or disadvantage, but I don't see a problem with the saves being modified in unusual circumstances. I'd just make sure that you're treating other players' unusual abilities with the same level of scrutiny.
P.S.: The Battle Master is, in part, 5e's version of the Warlord class from fourth edition. That class has the Martial power source, indicating that it's wholly nonmagical, with the following explanation:
You have become an expert in tactics through endless hours of training and practice, personal determination, and your own sheer physical toughness.
Best Answer
From the Errata:
The wording on this is a bit strange.
You could interpret it to be saying that instead of making a melee weapon attack, you can make an unarmed strike (inferring unarmed strikes are distinct from melee weapon attacks).
Or, you could interpret it to be that instead of making a melee weapon attack with a weapon, you can make one with an unarmed strike. This interpretation is a bit confusing, since it implies that your fists are not weapons, yet are capable of making weapon attacks.
You could make a case for either interpretation - the former plainly makes more sense as written, but the latter might be their attempt at enabling you to use your fists the way you would weapons for certain attack mechanics, while still not allowing you to do something silly like parry a hammer-blow with your skull, or enchant your feet to do +1 damage.