No, people don't just get to choose their facilities like that. They use the priority placement tool to create a list of the most in demand facilities. It attempts to calculate staffing based on current CPCs and projected CPCs (based on incoming personnel, outgoing personnel, and retirements, deaths, etc). Trying to figure out how it comes to some conclusions is hard, but just know that it is used in determining 'need'. Academy personnel will really only be assigned to level 7 and below terminal facilities (enroute people will get a list of centers in need [it could potentially be any center]). Occasionally, a level 8 facility might make it on the list (so far I have only seen Wichita and Oklahoma City). So, they go down the list and typically choose the next highest priority facility at a level 7 or below - some locations might be skipped if there is a training backlog. Also, they typically won't send more than 2 people to the same facility from the same class.
Here is a copy of the terminal placement list:
Terminal Placement list