The energies of the electrons in a metal are mainly given by the the band structure. In the free-electron approximation at $T=0$ the kinetic energies are only dependent on the electron density. These energies are a few eV, with highest velocities at the Fermi energy, of the order of 1 % of the speed of light. It is the ground state, zero point motion, there is no energy to lose.
At temperatures larger than zero, the electron energies are in thermal equilibrium with energies given by the Fermi-Dirac distribution. This only changes occupancies within a few $kT$ around the Fermi level (at room temperature, $kT \approx 25$ meV). Excited energies will lose energy, other electrons will get thermally excited. A dynamic thermal equilibrium. Average velocity is still zero, of course.
But when there is an electric field, a current and a drift velocity, there is a steady state, but no thermodynamic equilibrium. Electrons get accelerated during a relaxation time $\tau$, but then that energy is lost in interactions with the lattice - Joule heating