I have somewhat of a theory.
You remember this
thread I made? The three requirements mentioned by Sinead were: (1) skill, (2) reputation and the unknown (3) loyalty.
I realized that throughout the past two weeks I haven't run a single Shadow Mission. Not one. And I've applied for the position, and nada; I haven't been chosen to be a Royal Alchemist.
My theory is... what if doing more Shadow Missions gives you a better chance towards the position? For current Royal Alchemists, they really have no choice, as they are continually running missions and helping others on their missions to gain points to remain RAs.
And it might make somewhat of a sense... loyalty to the kingdom, as Sinead says, means fending off those Shadow baddies in the Shadow World and protecting the kingdom and whatnot.
It's very plausible IMO. Loyalty can't simply mean complete G9~G11 (and G12, when it comes out), as that has more to do with the Journal, and your "reputation."
Anywho, I'm getting too into this topic. But that's what I think. Completing Shadow Missions (be you a Royal Alchemist already or not) fulfills the "loyalty" requirement of the process. More Shadow Missions = better chances.
Just a theory though.