If the GM allows it, your friend could now use the alternative to the Beast Master ranger which appears in Tasha's Cauldron of Everything (TCoE, p.61):
Primal Companion, 3rd-level Beast Master feature, which replaces the Ranger's Companion freature.
This option has more synergy with class development - possibly mult-classing, too.
This alternative to the ranger sub-class does not rely on CR; it says:
You magically summon a primal beast, which draws strength from your bone with nature. [...] When you finish a long rest, you can summon a different primal beast. The new beast appears in an onoccupied space within 5 feet of you.
The Primal Companion alternative also provides three options to choose from for your companion, a beast of the land, of the sea or of the sky. Your ranger's companion will also evolve as your PC increases in ranger levels, proficiency bonus modifier and spell attack modifier - hence the synergy.
- Ranger Levels increases: Hit Points
- PB increases: AC, damage
- Spell attack modifier increases: to Hit bonus (attack modifier)
Not only that, you also get a way of returning the beast back to life:
If the beast has died within the last hour, you can use your action to touch it and expend a spell slot of 1st level or higher. The beast returns to life after 1 minute with all its hit points restored.
If your friend wishes to multi-class with a Rogue, it might be worth considering the sub-class of Mastermind (XGtE) at level 3, as he'd be able to use Master of Tactics and use the Help option as a bonus action to give his Primal Companion (beast) an advantage on attack rolls. This would mean that the Ranger would need to use one of its Attack actions so that the companion attacks. Still, it's a fair synergy. I would not multi-class until the ranger reaches 5th level though, as the Ranger will need the Extra Attack to increase their damage output and versatility.
In a word: Yes.
While it may be fairly well-accepted (though debatable) that a Beast Master Ranger is weaker than a Hunter Ranger, allowing the beast to attack freely swings things wildly in the other direction.
For the purposes of demonstration, I'll be using a basic Longbow Ranger, taking the Archery fighting style, starting with 16 dexterity, and taking +2 dex at level 4 and 8 to reach 20 dexterity. I'll be ignoring feats and other ranger builds to keep things simple, but they wouldn't really change the end result much anyways.
I'm using Damage per Round (DpR) as my measuring stick, taking into account chance to hit and chance to crit, using an average enemy AC at each level to get as practical a number as I can.
Basic Longbow Ranger
- Level 3: 5.5 DpR
- Level 5: 13.1 DpR
- Level 11: 16.6 DpR
- Level 20: 15.7 DpR
Wolf Animal Companion:
- Level 3: 5.2 DpR
- Level 5: 6.8 DpR
- Level 11: 16.1 DpR
- Level 20: 17.0 DpR
Longbow Ranger with Colossus Slayer:
- Level 3: 9.6 DpR
- Level 5: 18.4 DpR
- Level 11: 21.0 DpR
- Level 20: 20.0 DpR
Longbow Ranger plus Companion:
- Level 3: 10.7 DpR
- Level 5: 20.9 DpR
- Level 11: 32.7 DpR
- Level 20: 32.7 DpR
As you can see, while things start out OK at low levels, with the Ranger+Beast combo coming out only a bit ahead of the Colossus Slayer, this quickly changes once the Beast gets it's second attack. At this point, the ranger is now competing toe-to-toe with the better Fighter builds for damage potential.
Requiring the Beastmaster to use an action for the initial beast attack helps balance this out a bit, but ultimately it's still very powerful, and probably not wise to allow. As an interaction, it's just straight-up overpowered.
Best Answer
Technically, you have to use your action to tell your companion to do anything except move. From the PHB, page 93:
It says nothing about the animal companion continuing to perform the action you made it do in subsequent rounds. If you want it to keep attacking, you have to keep giving it orders. Effectively, it acts as a surrogate for you in combat. Note that, as pointed out in this answer, the Animal Companion isn't substantially weaker than the Ranger himself, and might actually be stronger, particularly at low levels.
I've seen the houserule that the Animal Companion should continue to attack an enemy once ordered to attack them once suggested quite frequently, and the DM is of course free to decide that this is the case. But this is still a houserule, not RAW.
As for "why wouldn't they just get a guard dog that would do just that after a few months of training", there is currently no explicit option that allows you to do this. You can buy beasts (including a mastiff) that are trained as mounts, but buying/training a dog to attack your enemies (which may include dragons, elementals, ghosts...) is entirely up to the DM. Even if they allow it, such a dog wouldn't get the bonuses that the Animal Companion gets and would therefore be weak at low levels and very quickly become effectively useless.