Certainly Anglicans were historically very hostile to Jesuits, but not because “Jesuits were founded to fight the Protestant Reformation.” That belief has been refuted by John O’Malley in his excellent book The First Jesuits (though it does make for a great joke about Dominicans vs. Jesuits. . . . ). Ignatius was barely aware of the Protestants. He founded the order originally to evangelize nominal Catholics (such as he had himself been before his conversion), and I believe missions to the heathen were also an early goal. Only when the Jesuits went into Germany and realized how many inroads Protestantism had made there did this become a major priority.
Then, in the late sixteenth century, the Jesuits became leaders in the effort to re-evangelize England for Catholicism. That’s how they became figures of fear and abhorrence not only to Anglicans but to English-speaking Protestants generally.
I doubt this has much to do with the attitude of modern American Anglicans, though. I haven’t myself seen a lot of this hostility, but it may exist in some quarters. The suggestion that Ignatian spirituality is too internal and introspective may be right. Episcopalians, and high-church Anglicans elsewhere, do tend to be suspicious of that kind of piety. This is both a reason why Anglicanism has been a spiritual refuge for me and a reason why it is spiritually unsatisfactory. I come from a Wesleyan Holiness background, a form of spirituality that has often been compared to that of the Jesuits (Wesley admired Loyola as “the best man who ever served so bad a cause” and was accused of being a Jesuit himself, although it was pro forma in the eighteenth century in England to accuse religious opponents of being Jesuits!).
Another possibility is that Ordinariate priests, who tend to be pretty conservative even by Catholic standards (especially from a liturgical point of view), may dislike the Jesuits because of their recent reputation for liberalism, and particularly for liturigcal laxity.
But the basic explanation may just be that Anglicans think of Catholicism primarily as patristic and medieval Catholicism. We tend to be colder to post-Reformation Catholic spirituality generally.
Edwin