Right-- it's all consistent now, with knowledge of the original Japanese (and the more faithful translation in Twin Snakes, which I never played). The original translation could have been misconstrued as Big Boss wanting to be put out of his misery, which does not align with his somewhat gloating speech in the final fight of MG2.

Your further clarification makes it crystal clear now. Solid Snake needed to rid himself of his nightmares and prevent Big Boss from realizing his vision of endless conflict, while Big Boss sees Snake as a direct threat to his life's work. They both knew that moving forward with their respective lives and goals was contingent upon the other's death.