Sperm and egg cells are “alive” in only a very limited sense. Neither can survive for any length of time in their natural environment, and neither can perform most of the normal functions of other human cells. They cannot receive nutrients or excrete waste, and they cannot perform cell division, which is the critical characteristic of life.
But once made whole by merging their two complementary parts, they follow an innate, complex program that, if they survive this first, most perilous journey to their natural but temporary home, connected intimately to their nurturing mother for the months needed to prepare them for the world we live in.
Neither sperm nor egg, by itself, could ever become what the two of them together will be, or do by itself what they are fully prepared to do together.
Actually, they can. Fructose is a component of semen. Eggs have a yolk sac.
Limited or not, egg and sperm have life.