The bottom line is that it should never be assumed that an internal standard will deliver better precision.
Agreed.
But an internal standard can be a very useful troubleshooting tool. Simply compare the RSD with and without the internal standard.
If the precision is
better with the IS, that suggests that the dominant error contributions to the IS and the analyte peaks were (highly) correlated. This implies that the "problem" occured when the peaks were still together: sample workup, dilution, injection, etc.
If the precision is
worse with the IS, that suggests that the dominant error contributions to the two peaks were (highly) uncorrelated, implying problem occurence during/after separation: loss on column, tailing, detector response, integration, etc.
If you are not using an IS, you can get the same information by comparing the areas of any two (comparable size) peaks. Simply look at the precision of the area ratio vs. the precision of the area.
That said, I concur with Bill's original suggestion, which amounts to a version of Occam's Razor ("simpler is better"); if the precision without an IS is adequate for your purpose, then an IS is unnecessary.