In practice, yes, unless the ranger's unable to command it.
Yes, the player of the Beast Master Ranger does effectively control his/her animal companion most of the time.
As you quoted:
The beast obeys your commands as best as it can.
It doesn't take an action unless you tell it to. It doesn't move unless you tell it to. It can use its reaction freely (e.g. for an opportunity attack), though it presumably would only attack a creature that you'd expressed hostility towards or vice versa, or one which is hostile to the animal companion itself.
In practice, this means that the Ranger player does indeed control what the animal companion does. There's no specific range restriction in which the companion can be commanded, but unlike Find Familiar and Find Steed, there's no telepathic bond between a Beast Master Ranger and their animal companion - so the only way to command the animal companion is verbally (or maybe with gestures, though I don't think there's any RAW description of that).
The 1st-level spell beast bond (EEPC, p. 150) does make telepathic communication possible:
You establish a telepathic link with one beast you touch that is friendly to you or charmed by you. The spell fails if the beast’s Intelligence is 4 or higher. Until the spell ends, the link is active while you and the beast are within line of sight of each other. Through the link, the beast can understand your telepathic messages to it, and it can telepathically communicate simple emotions and concepts back to you. While the link is active, the beast gains advantage on attack rolls against any creature within 5 feet of you that you can see.
It's a Touch-range concentration spell that lasts up to 10 minutes.
(Beast sense (PHB, p. 217) is another spell that essentially lets the animal companion function like a familiar; it's a 2nd-level concentration spell that lets you see and hear what a willing beast sees and hears, for up to an hour or until you use an action to stop doing so.)
In other words, the player controls the animal companion as long as it can hear the Ranger's verbal commands, or as long as it's within line of sight if the Ranger casts Beast Bond on it.
If the Ranger falls to 0 HP (or otherwise falls unconscious), Mike Mearls unofficially says he would rule as a DM that "I'd let it protect and aid ranger" - after all, the animal companion does have a special bond with the Ranger.
In addition, he replied to a question about what would happen to the animal companion if the Beast Master Ranger actually died as follows:
I'd let the companion stick around as a DM run NPC. Might remain friendly to the ranger's companions
Again, though, this is Mike Mearls' personal opinion as a DM, not an official declaration of the rules.
In summation: in play, the Ranger's player effectively controls the animal companion as long as the Ranger is able to somehow command the companion. The DM would likely control the animal companion in most other cases.
Channel Magic, external sources, and the optional facing rule
It is true that the most common reactions are unavailable to the homunculus servant. However, it still has some options.
Channel Magic is a reaction listed in the servant's stat block, functioning similarly to the spell delivery mechanism of find familiar. This is presumably the biggest reason reaction usage is specifically called out.
Channel Magic. The homunculus delivers a spell you cast that has a range of touch. The homunculus must be within 120 feet of you.
Eberron, Rising from the Last War p.62
Outside of the option inherently available to the homunculus servant, some magical effects and mundane situation call for the use of reactions. Spells, such as dissonant whispers and power word heal, can cause the target to use its reaction either voluntarily or forcefully. It is certainly possible within the rules for monster abilities, traps, or other situations to have similar stipulations. One example is the Path of Blades complex trap from Xanathar's Guide to Everything (p.118)
Magic items can grant reactions, as well. If you wanted to make your homunculus the envy of your friends by attuning it to an Ioun Stone of Greater Absorption (and your GM is permissive), well, that's an option you have.
Another niche option comes from the optional facing rule in the DMG (p.252), which allows creatures to change their facing as a reaction when other creatures move. This may or may not matter to the homunculus, but again, it is an option you have.
Best Answer
You must spend a bonus action each time.
You have to spend a bonus action to command the creature, but this command is limited to the standard actions and actions from the drake's stat block. "Kill these 3 goblins" is not an action the drake can take.
Since the actions the drake can take are all single use you have to spend a bonus action each turn to make it do anything else than take the dodge action. This is part of the balancing of the action economy of the feature.