Why was Dean so eager to die at the end of the Supernatural TV Show? Why not live a full life with Sam?

Dean was not “eager to die”. He resigned himself to die. Jack had said that he wouldn’t be involved directly anymore. He didn’t want Sam selling his soul or doing some kind of dangerous magic to bring him back because as he said, it never ends well. He had lived a proud life and got to see his little brother become a man. That’s not to say he wasn’t a man before, but Sam had matured and grown so much since he’d gotten him from Stanford. Dean could have lived out his days as a normal guy or grown old as a hunter, but he wouldn’t want to do so at any cost to the world or his brother.

He accepted his fate … dying on the job. He also knew that when they brought each other back, it ended badly. He wasn’t eager. He didn’t want to die. He just accepted it.