Visser Three was stationed on Earth before Elfangor and the other Andalites ever arrived. Deconstructing the canon events into a workable timeline supports this.
---Visser One arrived on Earth druing the time of the Gulf War, specifically Desert Storm, meaning early 1991, around January or February.
---It was mentioned Visser One spent 14-15 months on Earth. No more than 16 at the maximum. This places the time frame near March or April of 1992, with May as the longest cutoff date.
---It was mentioned in #10. The Android by Marco's dad that Eva had been acting weirdly for "a year" before she disappeared, which took place "two years" before the series started.
---The series started in 1996, but that's a debate for another time.
---Thus, Edriss landed on Earth in 1991, Essam died in 1992, she found Eva as a host in 1993, and a year later, in 1994, she faked her death to become Visser One, leaving Earth in the hands of Visser Three.
Now, my theory for why Visser Three came are several. Maybe he volunteered when Edriss left, thought that he could conquer humans single-handedly and become Visser One in the process. This actually fits his character, when you think about it.
Maybe there was an admissions process and his name was selected randomly, though I can't imagine that happening in something as structured and militarily ordered as the Yeerk Empire is.
Maybe he was assigned at a later date.
Maybe the Council thought, since he had given them information on Andalites that helped, he could do the same with humans, but this is questionably flawed since Edriss had done basically same thing with much greater success, although if this is true, it would mean the Council of Thirteen dropped the ball spectacularly.
Well, those are my opinions.