Suppose atoms of an ideal gas are represented by non overlapping wave function so that the system can be described classically. As time passes the packets spread. Therefore over a period of time we can expect the system to have overlapping wave functions requiring a quantum description. I need see why this has to be.
I remember doing that same question to a professor of mine (a long time agooooo), and his answer was that we could imagine that each collision is like an observation, and the wave collapses back into a localized packet. The answer was not very convincing, but in any case, remember that physics is not mathematics, and you can get very accurate physiocal results using some dubious math. But it works! In the end, you are correct, classical QM is not the ultimate theory, and a more correct way to do the calculation would be one based on quantum field theory. However this is beyond the scope of a QM textbook, and seems not to be necessary, because, in any case, the "approximation" (of assuming always localized packets) works pretty well. But rest sure that is you do a "first-principles" simulation (I am not sure you can do it analitically) using packets spreading as long as you want to become planar waves, you will also get the same result.
Let's make some order: you say "As time passes the packets spread". Why should they spread?
How do you imagine that these packets look like? Do you believe that initially the wave-packets are very small, and increase in width in time? No reason for that. As an example, assume that the gas consists in very light particles, and the gas is very cool, s.t. the wave-length of the movement of the particles is of the order of magnitude of the dimension of the box in which you keep the gas. Let the box be a cube of linear dimension L. Then, the uncertainty in the x, y, and z is L. There is no reason that this uncertainty change in time.
In short, if the description of the gas requires quantum treatment, then, at least from your explanation, describe them quantically from the beginning.
On the other hand if the classical description is possible (at ordinary temperatures such a description is good in many cases) then classical it remains. A gas in thermodynamic equilibrium, keeps its statistics of velocities constant, so, this picture with widening wave-packets has no reason.
Good luck !