Don't worry- I'm a Goku scholar, here to chime in.
Unfortunately with a multi-cultural icon such as Goku, you're going to have a ton of different iterations of his identity based on who is writing him at any given point.
I've seen both: versions of Goku that reject his Saiyan name in favor of his Earthling name, and versions of Goku that embrace his Saiyan heritage alongside his Earthling upbringing as two halves of a whole.
That being said: Goku's name is Goku. It is his identity.
There are plenty of trans people that are unbothered by deadnaming, misgendering, etc.
There are plenty of trans people who make exceptions or give concessions to specific people in their life because it's easier for them.
There is no reason to treat it differently. If Goku WERE offended to be referred to as "Kakarot", a name he hypothetically doesn't accept as part of his identity, then he would be well within his right.
Even as the show progresses, Vegeta starts saying Goku more, demonstrating his growing respect for Goku's identity as an Earthling. (Nevermind misremembered this part, the rest stands)
Goku doesn't have the same underlying trauma some trans people have regarding this identity discrepancy and how it makes society react to him. That being said, neither do all trans people.
Your reasoning for why they aren't the same is very shallow and reveals that you haven't considered this aspect of the topic deeply enough.
Edit: also I do want to point out that Goku's embracing of his Saiyan heritage, while always in the background as a story arc, really got a lot more play in recent years, and it seems that he has reached a point (in the Super anime) where he acknowledges that heritage without reservation.
/smugbrag
When I was younger, my former roommate's crush (and future girlfriend) described being part of our conversations as "like watching a TV show". How much of that was us being cool and unique versus how much was us autistically mirroring characters from pop media? Don't care, great compliment.