"Why doesn't the First Law (e.g., $\Delta U=W+Q$, where $U$ is internal energy, $W$ is work, and $Q$ is heat) incorporate bulk motion or momentum?"
For simplicity and convenience. We often assume that the system is motionless in our frame of reference, but this is not essential. I quote at length from Callen's Thermodynamics and an Introduction to Thermostatics:
In accepting the existence of a conserved macroscopic energy function
as the first postulate of thermodynamics, we anchor that postulate directly
in Noether's theorem and in the time-translation symmetry of physical
laws.
An astute reader will perhaps turn the symmetry argument around.
There are seven "first integrals of the motion" (as the conserved quantities
are known in mechanics). These seven conserved quantities are the energy,
the three components of linear momentum, and the three components of
the angular momentum; and they follow in parallel fashion from the
translation in "space-time" and from rotation. Why, then, does energy
appear to play a unique role in thermostatistics? Should not momentum
and angular momentum play parallel roles with the energy?
In fact, the energy is not unique in thermostatistics. The linear momentum and angular momentum play precisely parallel roles. The asymmetry
in our account of thermostatistics is a purely conventional one that obscures
the true nature of the subject.
We have followed the standard convention of restricting attention to
systems that are macroscopically stationary, in which case the momentum
and angular momentum arbitrarily are required to be zero and do not
appear in the analysis. But astrophysicists, who apply thermostatistics to
rotating galaxies, are quite familiar with a more complete form of thermostatistics. In that formulation the energy, linear momentum, and angular
momentum play fully analogous roles. [emph. added]
Callen then gives an example involving a stellar atmosphere in motion.
"Classically, why don't we see a difference in temperature for two identical systems of gas moving at different speeds?"
Because the temperature is defined to eliminate bulk motion; it measures the activity of the molecules relative to their center of mass, not the movement of the center of mass.
"How is this relevant to the piston–gas problem?"
Since we're free to assume any inertial frame regarding changes in temperature, let's assume for convenience that we're moving to the right at speed $u/2$, half the initial speed of the left piston. Now the problem is symmetric, with both pistons moving inward at the same speed. (The gas is now moving to the left at speed $u/2$, but this doesn't matter because we're going to focus on the gas temperature; as noted above, this is defined to be independent of the gas speed.)
Because the speeds are far less than the speed of sound in the gas, the pressure is approximately the same everywhere in the gas; this pressure decelerates the pistons to a stop, which occurs for both pistons at approximately the same time because of symmetry. At this moment, the pistons have zero kinetic energy in our frame, and this is the point where the gas achieves its highest temperature.
"So is the momentum of a moving region of gas ever relevant?"
Yes, if the bulk speed changes, which isn't the case in the frictionless piston–gas problem. This is the origin of the stagnation temperature, which measures the increase in the bulk temperature if the gas is brought to a halt. Here, the bulk kinetic energy of the gas has been converted entirely into thermal energy.
