Kings in the middle ages did impregnate servants BUT those servants were freqently of noble or “gentle” birth themselves. The daughters of knights, at the least.
Why mess with the runny-nosed maid scrubbing the privy floor when there were wall-to-wall ladies-in-waiting doing the less grubby jobs of helping the queen and princesses dress, doing their hair, mending their clothes, and serving their meals? Not to mention the daughters and sisters of the noblemen serving the king?
The children from those affairs didn’t become princes and princesses because they weren’t in the line of inheritance. But their mothers were often married off to someone with money and who wanted to be in the king’s favour. The king might acknowledge a daughter with an estate and minor title at some point to get her a better marriage or acknowledge a competent young son with a title, an estate and a military command.
In Norman tradition, it was common to give an illegitimate child the surname starting with “Fitz.” Giles Fitzgerald would be the illegitimate son of a nobleman named Gerald of Whatever, while Gerald’s legitimate son William would be William of Whatever. Fitzroy was reserved for the son of the king. There were lots of Fitzroys around sometimes.
William the Conqueror was illegitimate, but since his father, Robert, Duke of Normandy, had no legitimate sons, he picked William to be his heir. William of Normandy, later King William I of England, still got stuck with the name “William the Bastard” by his detractors.
Henry I of England had 25 known illegitimate children by several noble mistresses. He made one of his nine illegitimate sons the Earl of Gloucester and another the Earl of Cornwall. Others (mostly named Fitzroy) were given smaller estates and/or military commands, and at least one went into the church. And although Henry lost both his legitimate sons young, the earls of Cornwall and Gloucester backed up their legitimate sister Matilda’s claim to the throne. Of his daughters, one became an abbess and most of the rest married well — one was a countess, another a duchess, and another the queen of Scotland.
And what did the queens think of this? In the middle ages, they didn’t have much to say, unless they were Queen Regnant like Joanna of Castille, who, when her handsome husband, Philip Duke of Austria, messed with some of her ladies-in-waiting, cut off their hair and spread it across his pillow as a warning.