Inheritance laws are what make bastardy important. Simply put, bastards could not inherit. This sounds like a minor thing to us in the modern era, but it was a huge deal in former centuries. (Obviously, this has been since changed, and courts have struck such laws down in the modern era. See, e.g., Levy v. Louisiana, 391 U.S. 68 (1968))

In agrarian societies, the most common way to hold wealth and power was to own land and to pass that wealth down through the generations. That means that you have to create complex systems to ensure that wealth stays in the family. Families' biggest fears were that some nobody would parachute in and get access to the family inheritance. The principal way to deal with this problem was strategic arranged marriages. (Divorce, of course, was virtually unheard-of.)

You can see a fictional example of planned strategic marriage in the Jane Austen classic Pride and Prejudice, set during the 1810s. There, Lady Catherine de Bourgh-- the daughter of an earl-- has long planned for her daughter Anne to marry the very wealthy Mr. Darcy instead of the gentle-born but poor Eliza Bennet. The novel's eventual love marriage between Mr. Darcy and Eliza is exactly what holders of dynastic wealth sought to avoid.

Legal marriage was extremely tightly controlled and plotted over by families aiming to maintain their status and wealth. In many cases, this resulted in widespread incest among famous houses, including the Hapsburgs and Rothschilds. King Charles II the Bewitched of Spain (1661-1700) had only three grandparents, and every single one of his great-grandparents was descended from the same couple!

Now, to get to the stigma: there was one. Bastards were an unfortunate fact of life that the gentry and the nobility preferred to sweep under the rug. Legitimation of bastards made life much more complicated for everyone involved. Louis XIV of France (1638-1715) actually caused a political crisis by adding his bastard sons, the Duke of Maine and the Count of Toulouse, into the line of succession for the French kingship. In eras with relatively static class systems, enormous amounts of time and effort went into succession planning and inheritances, especially the kinds that would have political implications. Bastards threw a wrench into the works.