It’s almost like a different word, a hononym. To have and to have done something in the past. Neither being nor possessing really works for the “have done”. Being works for become because become has being as a part of its meaning as well as a transition from some previous thing that was before.
Though both are used similarly. I have ran. I am running. I will run. I guess have is still the odd one out since will is future tense for am. Though was also works. I was running. But was is more specific than have, it feels like “I was running” is a part of a narrative that includes a specific time, while “I have ran” doesn’t require anything else. It’s like you possess the previous action of running, so maybe it is apt. Language is funny.