I would normally understand that repeating an experiment many many times over a long period of time, and averaging the values, that should decrease the relative standard uncertainty with any measurement.
First, your understanding is, well, wrong.
This happy situation, where averaging decreases the type A uncertainty component, does happen only when the random measurement fluctuations can be modelled as white noise, that is, when they can be modelled as uncorrelated random variables.
Unfortunately, most of the noise processes that occur in experimental apparatuses during long-term measurements cannot be modelled as white noise. White noise is dominant for short-term measurements but, in the long run, noises with long-range correlations appear (e.g. flicker noise, random walk and, more generally, power-law processes).
For noise with long-range dependence (or long memory) it is no longer true that the statistical uncertainty decreases with time. For instance, it can remain constant, as with flicker noise or, actually, it can even increase, as with random walk. A good starting point to understand this problem is .
Typical noise sources of the above kind are the electronic noise of the measurement equipment, the seismic noise coming from the environment and the mechanical noise of the experimental apparatus (e.g. this is a well-known problem in gravitational wave detectors).
The consequence is that in any real experiment there is somehow an optimum measuring time (which, sometimes, can be quite short) at which the type A uncertainty reaches a minimum value and above which it no longer decreases.
In addition to this, in such kind of experiments, the ultimate measurement uncertainty does not depend on the type A uncertainty component but on the type B uncertainty component, that is, the uncertainty component which depends on the non-idealities of the experimental apparatus that lead to systematic errors. Examples are the inaccuracies in the determination of geometrical dimensions, the inhomogeneity of the test masses, etc.
For what concerns your idea, expressed in a comment, of averaging across different experiments, this is actually done. The recommended values of the fundamental constants given by CODATA that you can find in tables are actually averages across the state-of-the-art experiments. It should be noted, however, that some experiments can be affected by systematic errors with the same direction, and averaging would not reduce the effect. This probably actually happened in some of the past $G$ experiments (before the 1990s), because it was later found that the inelasticity of the fibre used in dynamic (time-of-swing) experiments could produce a systematic error .
 J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, 1994.
 K. Kuroda, "Does the time-of-swing method give a correct value of the Newtonian gravitational constant?", Phys. Rev. Lett., 75, 2796, 1995.