To undermine the legitimacy of stigmata would also undermine the legitimacy of the Church.
Let me explain…
Let’s say that Saint Francis, Saint Pio, or any other Saint with the stigmata was actually a charlatan. If these wounds were false “miracles” or supernatural phenomena based on lies (self-inflicted), one would rightly conclude that the Church’s authority is not longer reliable in determining Saintliness. I suppose the same could be said for other miracles as well.
That being said, the faithful are not obliged to believe in these miracles; however, I cannot possibly see how an individual would reject them, if they also acknowledge the authority of the Church.
Stigmata or other miracles do not legitimize the authority of the Church. However, if the Church officially, but falsely proclaims a miracle to be from God and it’s not, this proves the Church’s teaching authority is false.
Fortunately, this has never happened, that I’m aware of. The Church is right in being very cautious when it comes to these issues.