The ESPN 'campus' is an hour and a half from the Uconn campus and they have no presence on the campus. There are other schools that have better communications majors, and the presence of ESPN in the state doesn't really bring anything special to the educational options.
The advantage that Uconn alum have is that they are statewide celebrities from the moment they step on campus because CT does not have a professional sports franchise, football has never been as big a deal in New England as it is most other places in the country, and there are about 6 local newspapers that assign sports writers to the women's team throughout the season as well as all the local TV stations. On top off that, both NY and Boston are close enough that there are occasional reports carried in their various media. So the players become very media savvy and comfortable dealing with reporters and the whole process. Having a nationally available sports network as the broadcast partner doesn't hurt either. Add in the 11 championships in 21 years and the record winning streaks and the exposure the players get is significant.
In the list of the Uconn alum who have done broadcasting, you can add in Sue Bird by the way. But a lot of this is also based on availability - Skylar hasn't played in Europe so once the WNBA season is over she is available, Morgan is currently recovering from injury and not headed overseas, Sue has done college work when she has been rehabbing or last year when she took a year off, and Kara got her boost when she was available because she wasn't headed overseas. The broadcast companies want people they can work with and develop, and most women who are still playing professionally are just not available for prolonged stretches of time with their 12 month schedules. Edit - Chiney also became more available because of injury.
By the way - I think Kara is the best women's analysts out there, and one of the best analysts regardless of gender. I love listening to her, and when she and Lobo are working together, she raises the quality that Lobo brings as well.