That is a good point. And you did read everything correctly.
I just assumed everyone would want to play both at once. But maybe not.
I know that I do, personally, because three-player games are so s-l-o-o-o-o-o-w. I can't envision not being able to track two games, since each will be lucky to get two full turns per day. And one turn is more likely. I know that in my one game that's started, we've all been checking in pretty regularly, but are still only three cards into it.
TL;DR: You're right, and one game at a time is fine. However, I will heartily endorse at least trying two at one.