They used KENNEDY for about 5 years beginning in 1995 after S.L.E.P. much like they did LEXINGTON in the 1970's as a trainer for reservist. Still active, still commissioned, but reservist training mission. That in turn would have effected funding for yard maintenance as well. That had more to do with her 2001 readiness crisis than anything.
You are mistaken. The last dedicated training carrier was the Forrestal, which replaced the Lexington and as the link I provided earlier confirmed Kennedy deployed for both OIF and OEF in 2004. Was she neglected towards the end of her career? Yes. Did she continue to serve as an active fleet carrier up to and including her last deployment? Yes.