Find output impedance of BJT amplifier
It is a common assumption in hand analysis to assume the resistance looking into the emitter terminal is \$1/g_m\$ or in your notation \$r_e\$. If you follow this assumption, you will arrive very quickly at the same formula for \$Z_o\$.

A more complete model includes the effect of having finite source resistance. When looking into the emitter terminal, the source resistance appears to be reduced by a factor of \$\beta\$.

In your small-signal schematic you may note that regardless of whether \$R_C\$ is 0 \$\Omega\$  or 1 \$\text{k}\Omega\$ the current is dictated exclusively by the dependent current source.