In a word: No
(The Ranger would not be overpowered if their Animal Companion kept attacking once ordered.)
Companions must have a challenge rating 1/4 or lower. Many beasts that a player may choose are not capable of attacks of any consequence. Examples; cat, frog, bat, hawk, raven etc. only deal 1 damage point.
A player's choice to use a companion that is unthreatening may have more to do with their character's background or role-play interests. The companion being able to attack on its own brings more realism to the game as real animals would defend themselves if attacked and would not require a verbal command every 6 seconds. It also brings more consistency to the game world as NPC beasts act on their own.
WOTC acknowledged in the preface of Unearthed Arcana: Revised Ranger that the Ranger Archetype beast master has been unpopular to play as it is underpowered.
Many players want to play rangers, but few were happy with the class,
which held its place at the bottom of class power rankings by a
significant margin. The class’s individual features also filled the
top-tenlist of lowest-rated individual character features.
They attempted to rectify the issue with specific changes to the animal companion in the Revised Ranger archetype Beast Conclave allowing the companion to attack on its own.
The companion may take its own initiative and turn controlled by the player. The companion also gains the following features in addition to its own attack: at level 5, it may attack as a reaction to your attack; at level 11, it may attack multiple enemies during its attack on its own turn.
Beginning at 5th level, you and your animal companion form a more potent
fighting team. When you use the Attack action on your turn, if your companion
can see you, it can use its reaction to make a melee attack.
Storm of Claws and Fangs
At 11th level, your companion can use its action to make a melee attack
against each creature of its choice within 5 feet of it, with a separate
attack roll for each target.
Keep in mind that the base Revised Ranger class does not get an extra attack at level 5.
The beast cannot multi-attack.