Inconsistency in the calculation of the gain of an emitter follower
I think you just have an algebra mistake at the end. Starting from your last display equation: you can simplify the right-hand side to \$ (V_i - V_o) \frac{100}{120k} = \frac{V_i-V_o}{1.2k} \$. This is much bigger than the \$ \frac{V_i - V_o}{120k} \$ term on the left-hand side.

So you're left with \$ \frac{V_o}{0.5 k} = \frac{V_i - V_o}{1.2 k} \$, or more simply \$ 1.2 V_o = 0.5 (V_i - V_o) \$, giving \$V_o = 0.3 V_i \$. This agrees with the other analysis.