Emitter Follower Current Gain
Let me try a short - and more descriptively - explanation without formulas (which you already know): The input signal causes a signal output current Ic. Now it is important how this current is translated to voltage:

1.) Common emitter: There is a collector resistor which produces a corresponding output voltage Vc, which does NOT react back to the input. As a consequence, we can have a rather good voltage gain.

2.) Common Collector: Now, there is a emitter resistor which also "translates" the output current (forget that Ie is little larger than Ic) into a voltage Ve. However, this voltage strongly reacts back to the input because it is a part of the current-controlling quantity Vbe=Vb-Ve. More exact: It follows the input voltage at the base Vb - and, thus, does not allow any voltage amplification. This is the result of negative feedback.

(May I add the following - although I am aware that not all forum members are happy? The explanation under 2.) clearly shows that the BJT is a voltage-controlled device and that the base current Ib is not the controlling quantity. The working principle of an emitter follower cannot be explained using the current-control model.)