Here is the list of positions I have come up with where I feel the White Sox have sufficient depth to get through a season with no problems.
1. People named Reinsdorf.
2. The Grounds Crew. Thanks to Roger Bossard.
Right now I don't think they even have enough marketing people, they definitely don't have enough radio and TV people under contract, I'm not sure they have enough vendors considering they picketed the bloody stadium at the end of the season, and there's literally no position on the diamond where I would say "They have enough depth that they won't have any issue during this season."
So, would Elvis Andrus make sense for this team receiving an invitation to spring training? Absolutely. If they were to sign him, I would go so far as to say that between the injury histories of Moncada and Anderson, Leury's performance and injury history, and the inexperience of Sosa and Romy, there would be a high likelihood that he would make the team out of spring training. If he didn't' make the roster out of spring training we would probably need him for something by the end of May, and right now they have no one they could call up who could fill that kind of role. They would wind up calling up Burger and playing Leury or Romy full time at 2b or SS if someone struggled or got hurt, which is just a mess. The White Sox would be better with Elvis Andrus as an additional option than they are right now.
Elvis Andrus is not a starting player on anything other than a bottom of the league team. He is a guy who can be slightly above replacement level, we shouldn't count on anything more than that. But, the White Sox right now don't have replacement level infielders in their organization if anyone gets hurt or if anyone (including Leury, who was awful and who had a back injury last year) struggles again this year.
Now here's the problem. If Andrus was on this roster as a spring training invite, and the setup was for him to come up once someone had a chance to struggle through April or once someone got hurt, and he was in a spot where he might receive 200 plate appearances and play 50 games...that's not a terrible plan for $1 million. However, the question is whether we could limit him to that role. If Tony LaRussa was the manager, I would have to say "No you cannot sign him. If you sign him, he will receive 600 plate appearances regardless of how bad he is. It will not matter if a rookie can outperform him, he will get preference because he's a veteran."
Can Pedro Grifol manage a veteran and 2 rookies at infield positions? I don't know. I don't have a lot of confidence that they won't screw it up, but if they could do it correctly, having him in the organization could make them better overall.
It is also possible that someone could give an Elvis Andrus type player a big league contract. There are bad teams in the league, he could potentially start for one of them right now. The White sox should not offer Andrus that kind of deal, and if he would prefer that to a minor league deal with a team that has an outside chance of competing, then that's his business, find someone else who can fill that role.