The main rules for this mechanic are on my deviation
hereI know there is a few holes in my idea.
First off the fact that single Egg grouped Pokemon aren't included. My first thought is to add hidden Egg groups to those Pokemon, maybe like Monster for Rattata, that way they can act like every other Pokemon. Since the hidden group can't actually breed with the Pokemon, It'll clash with the Rare trait breeding. I figured since Rare Traits only have a chance then the secondary trait can be inherited every time the Rare one isn't.
Next is the how the Pokemon with Traits work when they breed. I think the males should be able to pass down the trait occasionally. Similar to how the Rare trait fathers can pass theirs, but with a significantly higher chance.
Another thing is I wanted to implement the traits into wild Pokemon. In Platinum you could find Pikachu in the Trophy Garden, with variants, you should be able to find Fairy Trait Pichu and Pikachu because you can find Roselia. Also in the Trophy Garden, since you can find Meowth and Eevee as daily Pokemon, on those days you can find Field Trait Pikachu and Pichu.
To go with the wild traits, the only instance I could find where two pokemon of matching egg roups were found together is Rhyhorn and the Nidos, and only in the Kanto Safari Zone. Meaning Wild Rare traits could possibly be more rare than shinies.
Edit<July 17th> My friend
Car1zardIsUrDaddy pointed out I hadn't mentioned genderless Pokemon. I figured I'd merge all three variants for them when they breed with Ditto, like half the time you get Trait A, the other half Trait B, but then sometimes you get a Rare. But After looking through the pokemon in the Genderless Egg Group, I've noticed only three of them are in a group other that Mineral, and that none of them really lend themselves to Variants. So maybe just disclude them like Legendaries.