Now I wont be focusing on Chilled in this discussion as he obviously doesn't have the mutant gene, which was shown with his battle with Bardock. However the characters for Dragon Ball heroes on the other hand are a different case. We don't know if there related to Freeza's family or not.
However I'm going to bite a bullet here and say that there not, or at the very least they don't have the mutant gene, evidenced by the fact that only Freeza and King Cold are the only two mutants.
Anyway with that out of the way. The topic for this post is disproving the idea that King Cold is incapable of transforming, let's begin.
To start off the debunking I'm first going to go through the appearances of Cold's children, grand children and a new face.
Now you may say that some of these designs are non canon, and to that I say that canon doesn't exist in this franchise. But the reason why I have them is that they are all designed by Akira Toriyama himself, so I imagine that the designs hold some weight. And as you can see, they all have the same consistent design theme through out all of them, they don't have any stripes, they lack those weird, brown, bumpy striped things that replace the gems and they all lack horns.
Now lets see King Cold's design.
It's almost as if King Cold looks beat by beat similar to every second suppression form in the franchise;
No really he actually looks beat by beat similar to every other second suppression form in the franchise.
And that ends the appearances section.
Now, I do admit that you can debate against the appearances you can argue that Freeza modeled his second form off his papa's true form, you can argue that Freeza and the rest of Cold's children and grand children are simply more mutated than he is;
But that's obviously not the case here, Freeza inherited King Cold's mutation, he is simply not more mutated than King Cold is. If King Cold is incapable of transforming than that would mean Freeza would be incapable of transforming.
And to cap off this post, the most likely reason for why King Cold never transformed, is because he completely and utterly underestimated the Super Saiyans.