Parmandur
Book-Friend, he/him
The design constraint of the beast master is to have a PC and their pet combined be roughly balanced with every other class/subclass. I think a lot of people make the mistake in assuming they want the pet to be basically another PC. Well, that would make the beastmaster too OP. Not gonna happen.
So what does that mean? The core ranger class offers many abilities regardless of subclass, so let's just compare the subclasses
Hunter:
At 3rd level, you can do around an extra 1d8 damage
At 7th level, you get conditional defense. Nothing too major. An AC bonus on subsequent attacks (not the first) from a creature against you. OR advantage on frighten saves. Or enforcing disadvantage on opportunity attacks. So basically, probably about 1/4 of all attacks against you if that.
At 11th level, you get more attacks, but not as big as people assume. You already have 2 attacks, but you're giving both of those up to either do volley or whirlwind. Again, conditional, and you're probably only getting a bonus of one or two additional attacks in those situations. So evened out, maybe an extra half attack per combat round.
At 15th level, you can basically half the damage against you once per round. (half to none on a made save, and half on a successful hit against you)
Now the beast master:
At 3rd level, you get your pet. Let's say a panther because Drizzt (yawn) 24 HP and 14 AC (increases with your prof bonus). What advantages does this give you (assuming your physical attack is better so not considering that a bonus to damage on a per round basis):
At 7th level, you basically can grant advantage any anyone via help at no cost to your normal attacks unless you dual wield (since it's a bonus action). Or it can dodge, which significantly improves it's damage reduction ability mentioned above since it is harder to hit and can suck up even more attacks that would otherwise be directed at you or an ally
- adv on perception checks.
- no cost movement (meaning you can use it to detect or attack creatures from a distance without putting your PC directly in harm's way, or attacking creatures you can't see from your current position, or granting an ally advantage via help at a distance--this is a big one)
- being able to knock opponents prone (again, at a distance)
- simply by being another person on the battlefield, can take attacks that would normally go at you or an ally. In mechanical terms, that's damage reduction to you or an ally. 24 points per long rest? That's significant. Even outside of all other benefits, which subclass ability grants you 24+ points of potential damage reduction at 3rd level?
At 11th level, it makes 2 attacks at the cost of one of yours. If each attack from you does 1d8+4 (ability mod)+2 (assume magic item or equivalent bonus at 11th level) points of damage, and each panther attack does +8 to hit and 1d6+6 damage, then compare: giving up 10.5 points to gain 19 points--or +8.5 points of damage per round.
At 15th level, when you cast as spell targeting you, you also target your pet. Like stoneskin, which reduces your damage by half, as well as your pet's.
So to compare:
3rd level: advantage beast master
7th level: advantage beast master (free advantage and damage reduction every round is better than highly conditional defense bonus from hunter)
11th level: advantage beast master. The 8.5 extra points of damage outpaces the extra half extra damage you get from volley or whirlwind. You'd have to attack 5 or more enemies on your turn for hunter to outpace this. How often does that happen when the beast master can do this every single turn?
15th level: advantage hunter. Both grant half damage (or other versatility with beast master), but hunter doesn't have to use a spell slot to do so.
In summary
The beast master and the benefits aren't nearly as bad as people keep assuming. In some cases, maximized damage in any given round might be less, but the versatility of the beast master far outpaces that. If you assume the pet should be the same as an equal level PC in terms of AC,HP,Dmg, of course it's gonna look weak. But for reasons already presented, that's incredibly flawed to look at it like that.
Quite correct: the problematic aspect is that this is a bit of a tougher Subclass to use effectively, but narratively it appeals to newer players. Hence, expectations and reality collide and result in a significant minority report of dissatisfaction.