Why do metals conduct electricity faster than heat?
If by electricity you mean the flow of current, then the question only makes sense (at least to me) under transient conditions with respect to how quickly steady heat flow is establish vs how quickly steady current flow is established, as discussed below. Under steady state conditions it's an apples and oranges comparison since current is the rate of charge transport and heat is the rate of energy transfer.
Transient Conditions:
Take a metal conductor, a wire. At time t=0 you apply a voltage difference between the the ends of the wire. An electric field is almost instantaneously established and electrons almost instantaneously begin moving throughout the conductor with some average drift velocity. So current is almost instantaneous throughout the conductor.
Take the same wire initially at room temperature throughout. The mobile electrons in the wire will have the same random thermal motion throughout the wire roughly proportional to the temperature. Now at time t=0 establish contact between one end of the wire with a high temperature constant temperature source with the other end in contact with a constant lower temperature equal to the room temperature. Thermally insulate the circumference of the wire to prevent heat transfer to the surrounding air.
The difference in temperature between the ends is analogous to the potential difference. The random thermal motion of the electrons near the high temperature end will increase. Through collisions with the electrons farther away from the hot end increased thermal motion will progress towards the lower temperature end until a linear temperature gradient is theoretically established along the length of the conductor. However, unlike the situation for current, this progression of thermal motion will not be instantaneous as in the case of the collective motion of charge. It will take time.
Steady State Conditions:
The applicable steady heat flow equation is
$$\dot Q=\frac {k_{t}A(T_{H}-T_{L})}{L}$$
The applicable current flow equation is
$$I=\frac {k_{e}A(V_{H}-V_{L})}{L}$$
The equations are roughly analogous with rate of heat transfer $\dot Q$ analogous to rate of charge transport $I$, thermal conductivity $k_{t}$ analogous to electrical conductivity $k_e$, and temperature difference $T_{H}-T_{L}$ analogous to potential difference $V_{H}-V_{L}$. The length and cross sectional area of the wire being $L$ and $A$, respectively.
But current and rate of heat transfer are different things, so it's comparing apples to oranges.
"However, unlike the situation for current, this progression of
thermal motion will not be instantaneous as in the case of the
collective motion of charge. It will take time." Can you please
elaborate on this a little, why this is, this is key to my question.
First, the motion of electrons in the case of current flow is collective motion, called drift velocity, which is proportional to current. That motion is due to the unidirectional electric force applied by the electric field to the electrons. That electric field travels in the conductor near the speed of light. So all the electrons immediately start moving.
On the other hand, the thermal motion of electrons when the conductor is heated is random. They don't collectively move along the conductor. Electrons with high thermal motion (random velocities) near the heat source collide with nearby electrons away from the source having less thermal motion (lower temperature) transferring kinetic energy to those electrons raising the temperature of conductor further from the source. They in turn collide with electrons nearby them and so forth until the thermal motion of all the electrons in the conductor has increased raising the temperature. All this takes time.
Check out the following video showing how the temperature of a heated conductor slowly increases along the length of the conductor as evidenced by color of the conductor.
https://www.youtube.com/watch?v=y-ptY0YG9RI
Hope this helps.
