Confusion in setting the bias point of an amplifier
You are correct. The emitter resistor, RE, is usually much smaller than the collector resistor so the voltage across RE will not change very much as the input signal changes. RE provides some negative feedback to help stabilize the amplifier. The output signal is really the voltage that appears across RC, and that is the resistor voltage that will change in proportion to changes in the input voltage. In general, you want to bias the amplifier so that the voltage across RC is about half of the supply voltage.