But here's where it gets twisted:

Of course Saint Olga's Academy originated from Mewni. After all it's Head Mistress, Miss Heinous, is implied to be part of the royal family, even though she tries to hide it. The real puzzle here is, why hide that? Why Does miss Heinous feel the need to hide her identity as part of the "House of Butterfly" and feel the need brainwash herself into becoming a perfect princess?

As we have seen in Mewnipendence Day, The royal family's ancestors once participated in what was basically a genocide against the monsters in order to establish the kingdom. When Star realized this, she found it to be upsetting. Miss Heinous, being part of Mewni's Royal family and much older than what we've seen of the members of the House of Butterfly, is possibly aware of the real story behind Mewnipendence Day and the reality of the Monster Massacre. Since someone like Star, an independent and non-brainwashed princess, found this to be upsetting, then chances are that the non-brainwashed and non-repressed Miss Heinous (if that's her real name) would find such facts to be upsetting too (and might be more emotionally worse off by being there to see it happen). Miss Heinous probably brainwashes herself and runs Saint Olga as a form of denial. She does it to deny being part of the "House of Butterfly" (perpetrators of old war crimes), so she can also deny her knowledge of the Monster Massacre (and maybe her participation) and probably deny how horrible it makes her feel.

As for the continued operation of Saint Olga's, this is both Miss Heinous and the Kingdom of Mewni's doing. By continuing to brainwash princesses and crushing their spirits and personalities, they eliminate sympathy. And, by eliminating sympathy in favor of soulless duty to tradition, Saint Olga's princesses are trained to not care or think too deeply about their place in life, their actions, or what is going on in their respective Kingdoms. By doing this the Kingdom of Mewni is able to continue it's operations against the monsters, with nothing in the way and no one protesting the mistreatment of the monsters. I suspect that no one (with one exception), not even Mewni, remembers or understands that the Kingdom of Mewni, the "House of Butterfly", and Saint Olga's are deeply related to each other.

   On the question of why other Kingdoms participate in the St. Olga’s program despite this being a Mewni institution, there are two possibilities:

1) Other Kingdoms have a similar story to Mewni's when it comes to monsters or their enemies, so wish to protect their reputation and legacies.

And/Or

2) Mewni's St. Olga's program has taken a life of it's own, and once again the promise of protecting a reputation and legacy made other Royal families want to participate. As a result, the murderous legacy of the Mewni Royal Family has begun to damage everything and everyone else around them.