The "wear and tear" is minimal; no more than running the engine at the same RPM in steady state or acceleration (and as noted, arguably less). The majority of the energy is converted to heat, and is easily dissipated via the cooling system, as is the design.
There is a solid argument for clutches being designed for one-way power transmission, but this is largely negligible as well, as vehicles have, for the last 90-or-so years, been designed to allow engine braking (as long as we've had to negotiate cargo down mountain passes).
As for saving wear and tear on the brakes, in modern cars (i.e. since the advent of disc brakes in ca. 1950's) there is nothing to be gained there. In cars with 4-wheel drums, however, you've got a race. Service brakes are designed for short bursts of heavy heat dissipation, unlike engines, which are designed for long-term small bursts of heat dissipation. When the braking situation calls for long-term but reduced magnitude conversion of kinetic energy to heat energy (such as trundling down the far side of the Rockies) the engine is better. But in daily use, in Texas, especially Dallas, there's no point.
The better point, though, is that there is no point arguing. If you enjoy engine braking, have at it. If you don't, don't do it. Hardly matters...
~an engine braking advocate
PS: JakeBrakes are a different animal, but the same principles of heat dissipation applies. The major difference is that Jake Brakes dissipate more heat through the exhaust owing to the double-pumping evacuation of the combustion chamber thereto.
PPS: even though modern automobiles are capable, in general, of not overheating their brakes on mountain passes, as @Ian_Jaeger showed, sometimes they still do; thus, among other reasons I advocate strongly the engine braking technique for long/steep downhills.
PPPS: @Owen_Soccer22's reverse thing is a total myth, unless you have regen...