Jesuits are just doing crusades but in a more psychological and cunning ways... And I say this from broad experience and insider knowledge.
The queen of England did not send ship loads of gold to do charity work in the colonies.
The native tax payers to this day pay the salaries including the priests who work as principals or head masters.
The land is almost always ill gotten, by force or on 99 year lease for $1.
And in many parts of the world these schools were started explicitly to breed subjects loyal to the colonizers, Macauly's children.
One thing I do know is that their schooling helped them to get jobs and get out of poverty when the software revolution happened.
I am not well versed in European history but common sense tells me a conversion collusion between the protestant British Queen and Catholic Jesuits has more probability to be a fiction.
Their contribution to the actual school is 0. In fact it legitimizes their continuing hold of land and wealth grabbed from the natives and temples for a "good cause"
Colleges like Loyola in Chennai built on temple land manufacture derascinated Hindu hating Indians who go on to work in media, bureaucracy and judiciary and it does have a substantial negative impact, especially in a country trying to decolonise.