Why does the common collector specifically have a high input impedance?
Quite right, but consider how transistors must be biased...
Consider comparing a common-emitter stage with a common-collector stage. Both have the same magnitude of gain (common-emitter has a gain of -1, while common-collector has a gain of +1).







simulate this circuit &ndash; Schematic created using CircuitLab

It is possible to use a large \$R_E\$ in the CC stage...considerably larger than the \$R_E\$ of the CE stage. You must leave lots of biasing headroom in the CE stage for the collector to swing. If the CE gain is large, the collector-swing headroom consumes almost all the DC supply voltage.... that makes RE1 much smaller than Rc.