Genetics empirically have a role. It's not pure speculation when we see that it's hereditary:
- Garp & Dragon
- Dragon & Luffy
- Roger & Ace
- Linlin & Katakuri
All top tiers that are known to have borne children had at least one child with COC. Oda doesn't actually need to tell us that it's hereditary. There is very strong empirical evidence that it's hereditary.
Btw, Garp is not confirmed to have CoC. In fact, he likely does
not have CoC because the Vivre Card went out of its way to specify Sengoku having CoC, but never said a thing about Garp. Which makes sense, Garp doesn't really have much quality of a conqueror.
Saying that just because some daddies and children had CoC in common, it means it's genetics doesn't make any sense. Every single example you gave have a very specific personality trait which runs in all of them- and canonically, it has been stated over and over and over again that these personality and innate traits are the reason why people get CoC, not genetics.
In fact, something like genetics seems like an ehhhh way of getting something that has been hyped as "a power those with traits of conquerors get".
Genetics don't decide these stuff.
No, many Conquerors only had "desire to be free" as their dream.
Kaido did not have some strong desire to be free. Neither did BM. Neither did Zoro.
This is about as potent as saying "Kaido has a desire to die honorable death. So someone else who has desire to die honorable death must have CoC"
Or "WB wants family, so everyone who wants family has CoC"
These are just secondary ambitions and stuff. It doesn't take anything away from- nor does it provide anything to their innate traits and qualities of conqueror. And,
again, it has already been canonically stated over and over again what makes a conqueror haki user.
What was Ace's dream that made him a Conqueror? At Eight?
Ace always had this desire to stay above others- the competitive spirit that he has to be stronger, we see it very often. Other than that, Ace was the eldest brother- the responsible one, the leader. He had it all; he had all the qualities you required for it.
In fact, even the scene where it got triggered only highlighted his role as a leader; he was responsible for not only his little brother's life, but also Dadan's (iirc on latter), and in a desperate situation with the responsibility of protecting others on his shoulders like the elder brother and leader he was, it finally went active from dormant.
Regardless, Ace's dream of this or that did not change his innate disposition as a leader (which he has ALWAYS been, even before Luffy came he was the one who took care of Dadan's family and brought them food), and his competitive spirit which was always high.
No, her ambition is to be free:
She idolises Oden because Oden was free. Saving Momonosuke is not her ambition. Luffy asked her to save Momonosuke and Momo is an important piece for the end of the world.
She has the resolve to stand up to the strongest in the world and fight for her freedom:
Yeah, I did say it's freedom and Oden stuff. But denying it's Oden stuff as well is wrong- she is STRONGLY trying to fulfill Oden's will, looking after Momo and risking her life for it.
I mean, c'mon, it's someone who literally identifies as Oden and said so multiple times. We can't deny she's obsessed with him in more ways than one lol. It becomes a character flaw when a character is more someone else than themselves.
And that's not a bad thing. Flaws mean there is a lot of room for character development- like, she's definitely ending this arc as her own individual instead of this pathetic Oden persona she took on.
But yeah, strongly following someone else to the degree she is doing is nothing like a Conqueror. It's quite opposite.