Let me clarify. You can do whatever charts/plots you want and add spec limits. Just not control charts. Control charts are designed to answer specific questions.
1. Which components of variation have the greatest leverage? Ultimately you want to use them to compare sources of variation. Where should you work to improve the process? This is the purpose of the X-bar chart. The spec limits are a function of the within subgroup variation (A2R-bar). The variation in the dots plotted against the control limits are representative of the between subgroup sources of variation (a function of the x's varying at the sampling frequency). In fact, the dots are biased to the between sources because the within sources are averaged. The X-bar chart does NOT assess stability (neither does the Individual chart). It can't as it is a comparison chart. You can't assess stability of one component by comparing it another component, you must compare the same component to itself over time. Shewhart was brilliant.
2. Is the basis for comparison consistent/stable? Before you can make the comparison, however, you must make sure the basis of comparison is consistent/stable. This is the purpose of range and moving range charts. If the basis is unstable or inconsistent, what good is the comparison?
"All models are wrong, some are useful" G.E.P. Box