The main complaint against them - often used simultaneously by the same person - is that they aren’t animal enough, and/or that they following the main character too blindly.
Animals are extraordinarily generous creatures. Most dogs get left at home while their owners work. The lucky ones are ‘car dogs’ - like my father’s dog, who rides around with him but has to stay in the car while Dad does this work - or farm dogs - the latter usually having jobs with require them to work for their humans.
Horses spend most of their lives locked in small stalls or pens, only to come out when humans require work for them. Humans give them horribly conflicting signals and frequently cause them pain, and the average horse still does their best to do whatever’s required. All a horse takes to be happy is a human who treats them with a little respect and praises them for doing what they were asked to do.
These animals do it for a little affection and some food, right?
Animals go through a process we like to call training, but which really amounts to systematic brainwashing. Horses as an example:
We take them from their mothers and put ropes on their heads while they’re small. Then we teach them that it’s uncomfortable to fight us, and comfortable to go along with us. We teach them that there is no freedom beyond what we grant them, and that good things come from us. We make them want to do their bests for us.
The same can be done to a person. If they’ve been raised to believe that life revolves around another person, and that person treats them reasonably well, they’re unlikely to go off on their own and take up their own thing.
My point then: it’s incredibly animal-like for a telepathic familiar to continue following a human who doesn’t treat them as an equal, and to continue loving said person. Further, even a speaking animal (most people assume this makes them smarter - ehm, no) is likely to continue with this mindset.
So rather than putting attention into making telepathic animals more independent, how about thinking of the way it reflects on a character that they’re willing to keep something they communicate intelligently with under a form of slavery? How about concentrating on making a telepathic animal have none-human intelligence, rather than simply being smart-like-a-person?