I do not completely understand the intuition behind equivalent sources used in source transformations
There is an unstated assumption that the circuit is linear. It's easy to devise a nonlinear circuit that would look the same to any real value of resistor but fail when certain voltage sources were connected. Think of a series battery + resistor with a reverse-biased diode across the terminals.

Since the circuit is linear, the superposition theorem applies. That (and the uniqueness theorem) is adequate to prove Thevenin's theorem. It also applies more generally with resistors replaced by impedances. 

Since it's linear it only has to behave the same with two values of resistor attached to be identical for all possible applied voltages to the two terminals (ignoring the degenerate case where the voltage is zero for both resistor values).