I was also surprised to read that heat is proportional to the current. In addition, increasing the temperature also increases the resistance in the conductor (Cu). It's around 0.4% per 1C for Cu, around 20% more heat at 70C.
Not sure about US, yet some high current lanes (thinks of threephase ~400V x 36A; IEC 60502-1) in the households are actually made of Al, not Cu. They tend to be underground though, the wires in the walls are still Cu.
Silver wire is used for some things, but it is a lot more expensive than copper which rules it out for most applications.
Here is a table of the conductivity (in units of 10^7 S/m), the density, and the cost of copper (Cu), aluminum (Al), silver (Ag), and gold (Au).
Cu Al Ag Au
Conductivity 5.96 3.5 6.3 4.1
g/cm^3 8.96 2.6 10.5 19.3
$/kg 9.03 1.2 1030 92100
If we had a copper wire with a specified capacity in amps, here is what aluminum, silver, and gold wires of the same length and capacity would weigh and cost as a percentage of the weight and cost of the copper wire, and what their diameter would be as a percentage of the diameter of the copper wire.
Weight Cost Diameter
Al 49 7 139
Ag 110 12646 97
Au 310 3190000 121
Not sure about US, yet some high current lanes (thinks of threephase ~400V x 36A; IEC 60502-1) in the households are actually made of Al, not Cu. They tend to be underground though, the wires in the walls are still Cu.