As long as there's an option for a pet designed to be in melee without that crippling the master...
Neither master nor beast must be more restricted in their action economy than if the PC just went out and bought a war dog. Yet the design must cater to those that don't want a disposable pet you replace all the time.
In short, the design can't be balanced since while ½ + ½ = 1, the master being half a character is unacceptable and the pet being only half as strong as another party melee:er doesn't work.
At the very least, we're talking ¾ + ¾ = 1,5. That is, don't even try to make a Beastmaster subclass balanced.
Just slap a sidebar explaining that this character option is by necessity stronger than the baseline, and requires the DM and the group's explicit agreement to be used.
Anything less and we'll just end up with another compromised designed destined for the scrap heap...
I agree with your logic about a beastmaster can't be less than a normal character plus a purchased pet. But a purchased pet, once you hit 5th levels, is like 10% or less of a normal character. And that gets less and less as you level.
So I wouldn't see the balance point at 1.5, I'd see it at 1.1. Near enough the other classes that it's still on the same curve, just in the front side of it.
I don't like the action economy stuff, and how the PHB Beastmaster is done is quite bad even after the errata. But from a gamist perspective, I can see having a characte that is viable on their own but gives up some of during play to make their beast more viable so that together they don't overwhelm the other classes would be
one workable design. Now, that starts with viable so giving up your action to have it attack isn't on the table, and there are likely ways outside action economy that fit the narrative better - I'm open for that.
Just to brainstorm, what about a ranger that starts with the idea of character + purchased pet? In other words, a ranger who doesn't get a pet but enhances what they have. It wouldn't even need to be a subclass - we already have Hunter's Mark as what should be a feature that they reused the spell mechanics for, we could do the same. Several until-end-of-long-rest spells that can buff a pet, which upcast to keep the pet on level, and some bonus action spells that could be used during combat. So a ranger keeps all of their martial goodness, trading the majority of their casting for a pet. And if there is no pet for whatever reason, they have their casting back (starting the next day).