Correct. And most races are qualifying races, as if they have subraces, you can exchange the subrace for a Genasi subrace to make them qualifying. Basically, everyone has to have a connection to the elements in some way. Either they're born with it, which is a qualifying race, or they get it later, which is a qualifying class.
And multiclassing will be encouraged and fixed. I've already mentioned that ASi's are based on Character level, not class level, with fighters getting their extra ones at the appropriate levels. I've toyed with the idea of doing other things to make muliticlassing more beneficial, but I think that may be enough, honestly. Most people don't like multiclassing because of the lost ASI's.
The only limit is that, if you're going to take a Qualifying Class as your qualifier instead of a race, and then multiclass, you have to take at least three levels in that class to qualify. So, three druid, monk, or Sorcerer levels, then you can do whatever, and those have to be the first three levels. That gives you a strong enough connection to the elements to be chosen for these missions.
This does mean that normal human and variant human can only be included if they're druid, monk or sorcerer. Humans with Genasi subrace just use the Genasi stats.