Has IUPAC been inaccurate in their 1994 definition of kinetic stationary state? Summary of IUPAC definition
$\def\d{\mathrm{d}}$In IUPAC recommendations from 1994[1, 1166–1167], the authors discuss the process
$$\ce{A &lt;--&gt;[$k_1$][$k_{-1}$] X\\ X + C -&gt;[$k_2$] D}$$
A kinetic stationary state or steady state is defined very reasonably:
$$\frac{\d\ce{[X]}}{\d t}=0.\tag2$$
They go on to emphasise that condition $(2)$ is not equivalent to $\ce{[X]}$ being constant, 'not even approximately'; simply that the change in $\ce{[X]}$ is small compared to changes in concentrations of $\ce{[A]}$ and $\ce{[D]}$ because $\ce{[X]}$ itself is small. If $\ce{[X]}$ were constant in a stationary state, they claim it would lead to the following contradiction.

Say reactant $\ce{C}$ is in excess. Assume that $\ce{[X]}$ is constant. Then the law of mass action $v = k_2\ce{[X][C]}$ would reduce to some $v = k'\ce{[C]} \overset{\mathrm{excess}}{\approx} k\in\Bbb{R}$. In other words, production of $\ce{D}$ could continue at a constant rate even after $\ce{A}$
has run out.

The document concludes that $\ce{[X]}$ cannot possibly be constant in steady-state conditions.
How I would resolve the apparent contradiction
I do not see a way around $\ce{[X]}$ being constant. Indeed, for any reasonable function[a] $\ce{[X]}:t'\to \Bbb{R^{0+}}$ equation $(2)$ implies
$$\int_0^t \frac{\d\ce{[X]}(t')}{\d (t')}\d(t')= \ce{[X]}(t) = r \in \Bbb{R^{0+}}\tag3$$
where $t'$ was simply a dummy variable.

Resolution to contradiction: If $\ce{C}$ is in a large excess, $\ce{[X]}$ is small, and $\ce{A}$ has run out, it is simply not possible for steady-state conditions to hold. In other words, it is our stationary state approximation $(2)$ which fails to hold for $\ce{X}$. So we can easily keep condition (3) for a steady state.
More accurately, if $\ce{A}$ has run out, the only possible stationary state would have been $\ce{[X]} = 0$. This is because, at a stationary state,
$$\ce{[X]} = \frac{k_1\ce{[A]}}{k_{-1} + k_2\ce{[C]}} \overset{\ce{[A]} = 0}{=} 0.$$
If $\ce{[X]} = 0$, it is unfeasible for the reaction to continue at non-zero constant velocity since $v = k'\ce{[C]} = 0$ (where $k' = k_2\ce{[X]}$).


Question

Is / is not the concentration of $\ce{X}$ constant, approximately or otherwise, in a kinetic stationary state (steady state)?

IUPAC states $\ce{[X](steady\ state)}$ is not even approximately constant because it would lead to contradictions.
I claim that $\ce{[X](steady\ state)}$ is constant, and contradictions surface when a steady state itself is impossible given certain constraints (or the constraints themselves are contradictory).

[a] Among other things that it is diffentiable everywhere in its domain (i.e., reaction rate is defined for every time $t'$).
[1] Muller, P. Glossary of Terms Used in Physical Organic Chemistry (IUPAC Recommendations 1994). Pure and Applied Chemistry 2009, 66 (5), 1077–1184. DOI: 10.1351/pac199466051077. pages 1166–1167
IUPAC should say they are defining the &quot;steady state approximation&quot; rather than the steady state.
There are two approximations in this technique.
First, that final reaction step is not an equilibrium (reverse reaction rate is zero).
Second, that the intermediate is so unstable that its concentration (and therefore the rate of change of the concentration) is approximately zero.
Compare the IUPAC definition to the following 1954 U.S. National Research Council explanation:

STEADY STATE APPROXIMATION
The most generally applicable of these simple methods is the so-called &quot;steady-state approximation.&quot; A steady state may be defined as a condition in which the rates of change of the concentrations of the several intermediates are very small compared to the rates of change of the concentrations of the reactants and products. This condition is realizable whenever the ratio of the concentrations of the intermediates to the concentrations of the reactants is very much less than unity. When this condition is not attained, the method is not applicable; however, it should not then be necessary since the (larger) concentrations of the intermediates could be measured by experimental means. ... The steady-state approximation consists in setting the rates of change of each of the intermediates equal to zero and in solving simultaneously the resulting algebraic equations.

Overall, if the situation was truly, not just approximately, steady state, yes the concentration of the intermediate would be constant, but the point is that the &quot;stead state approximation&quot; is useful even when the concentration of the intermediate, though always near zero in absolute terms, decreases to say half of its original concentration over the course of an experiment (as the starting material, A, decreases to half its original concentration).
Additionally, a very small $\ce{[X]}$ does not purely-mathematically imply a small rate of change of $\ce{[X]}$, because $\ce{[X]}$ could rapidly oscillate within a small, near-zero, range, resulting in brief instances of large rate of change.  Instead, $\ce{[X]}$ being small over a given period of time really just places a limit on how long of a time the rate of change can exceed a certain value.  For example, if $\ce{[X]} &lt; 0.001\ \mathrm{M}$,
$|\mathrm{d}[\ce{X}]/\mathrm{d}t|$ must not exceed $1\ \mathrm{M/s}$ for more than $0.001\ \mathrm{s}$.