Money and winning are what attracts. The Sox usually come up short with both. It would be very unusual for a major free agent to take significantly less money to come to a team with the White Sox recent history.
MLB players love Chicago. They couldn't care less about Bridgeport. They just drive there 81 times a year. They don't live there. I'm pretty sure most Sox players live in those buildings built on that old par 3 golf course by Randolph and Wacker and Columbus.
I'm sure they would prefer to play in a full stadium, but the amount of their paycheck trumps that. And there is no problem with Guaranteed Rate Field. The playing surface is supreme, the clubhouse new, and other amenities, modern. Unless it's a flyball pitcher, something the Sox probably wouldn't bother to pursue, the stadium is not an issue.
And endorsement money with MLB players pales in comparision to those in other sports. Yet, in Chicago, even with the Sox, you probably have a chance to make more money off the field than many other places.