> what science is at the core (i.e. making repeatable observations to establish fact)
These "facts" are predictions about what will happen in the future.
Physics tells us what will happen to physical systems, e.g. our understanding of gravity tells us what will happen to a system of masses in the future.
Chemistry tells us what will happen to chemical systems, e.g. our understanding of reactions tells us what will happen to interacting chemicals.
Medical research tells us what will happen to your body and disease when some agent is introduced to the system.
Climate science tells us what will happen to the climate, and it is based on the same concepts as the most fundamental and rigorous science: thermodynamics. If more heat enters than leaves a system, the temperature will increase. There is nothing unscientific here.