Which variable dictates the core temperature of any star almost entirely?

Initial mass

The primary factor governing how hot a star's core becomes is its initial mass. Gravity acts as the sculpting force, resulting in a greater inward gravitational pull for more massive stars. To counteract this crushing force and prevent immediate gravitational collapse into an infinitely small point, the core must generate an equal and opposite outward pressure. This necessary outward pressure is achieved only by significantly accelerating the rate of nuclear fusion occurring in the core, which directly results in substantially higher core temperatures. This dependency on mass is what fundamentally separates lower-mass, long-lived stars from high-mass, short-lived giants.