In the times and places we are discussing not everyone who was elite was educated and not everyone who was educated was elite. Nor was fencing education universal among the educated elites. Much like today, you can go to an elite school and not learn to fence and you do need to have an elite education to learn to fence. That doesn't mean fencing isn't an activity which is associated with high status, it is, I agree with you, but much like in today's world Polo is an activity associated with high status, but not everyone from elite circles does Polo.
> Learning to fence was traditionally very much part of an upper-class education.
I think this is more true about the 18-19th century than the 16th. As what constitutes an upper-class education was still developing at this point and often scholars just paid professors to teach them a subject they thought would be valuable.
Thibault, a Destreza master (1574–1629), was a wool merchant i.e., middle-class not elite. He learned Destreza while traveling. It was only after he had achieved fame and glory from his Destreza that he went to university and studied Mathematics. It is unknown how much formal education he had before this.
Fencing texts after the invention of printing press start looking more and more like instructional on how to teach yourself to fence. This is because there is a Middle Class which is buying fencing books to learn to fence. Some of this is because fencing and certain other martial arts are associated with class status. Some of this is that the Middle Class as an upward mobile class is obsessed with education and demonstrations of skills associated with the mind.
Fencing schools at this time period were not always run by elites and while generally most people with money and free time attended, money and free time an elite does not make:
"As part of the public amusements for the establishment of the House of correction in 1596, (who would get a part of the entrance fees), the town council included ‘comedians, fencers and other plays or novelties, which could by seen for money’. According to Pontanus, the school was located next tot the guild room of ‘d’Eglantine’, on the top floor of the SaintMarguerite Church, which at that time served as a small meat hall. When Hooft wrote his Warenar in 1616, he named the fencing school as one of the preferred places of the ‘gilded youth’, after taverns, the dance room, the kaatsbaan (sort of ballgame - n.o.t.) and the librij’ (the city’s library in the new church)." [0]
Next to the guild house is about as middle class a location as you can get. The rich kids hang out there, but probably the kids of wealth merchants not lordlings. Aristocratic warriors like knights (although at this point we are starting to approach the end of the Knightly class as a class of warriors), generally did not attend fencing classes as they would be tutored and/or taught by family.
> Also, fencing is about much more than hurting people: the very word comes from the word "defense" which doesn't necessarily require hurting somebody else.
Defending yourself in a sword fight i.e., two people locked in a lethal struggle using edged weapons, is a situation in which most fencing techniques would involve hurting the person who is attacking you. Just because an action is defensive does not mean it is not violent. Most fencing in their period isn't wholly focused on self-defense, it is focused on the activity of fencing; which includes training to win renown in the Salle or the duel.
[0]: GERARD THIBAULT AND HIS ‘ACADEMIE DE L’ESPÉE’by Prof. Mr. H. De La Vontaine Verwey (2012) https://signoresimone.files.wordpress.com/2014/10/gerard-thi...