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My idea for breeding:
Characteristic and natures play a role on breeding now. Not only just egg groups though those still apply.
Pokemon of the same egg group can still breed with one another but depending on the Pokemons characteristics the egg frequency and the hatched Pokemon can vary.
For example if the Pokemon have compatible characteristics then the frequency of eggs is increased. If they don't, than eggs take longer. If the natures are compatible the more likely the offspring will have better stats, if they don't the stats will be poor. If both Pokemon are 100% compatible in every aspect the likelihood of shiny Pokemon goes up dramatically with every egg.
Maybe even add a feature where your Pokemon can grow tighter bonds with one another if they battle together. This bond can correspond to the increased odds of better offspring.