Lutherans, for example, don’t run around proclaiming they are “saved”, yet they believe in Sola Fide.
Meeting Baptists in college came as a shock to me before my conversion. Their POV struck me as a bit arrogant.
It struck you as arrogant probably because you reject salvation by grace through faith. If I believed that my good deeds and works of righteousness merited salvation then, of course, someone who came along and said they "knew" they were saved would be appalling. It would be like they were saying they were good enough to be saved. But we aren't saved because of our righteousness - all our righteousnesses are as filthy rags when deemed "good enough" to earn eternal life. Our sin debt MUST be paid. We need a redeemer - one who pays our ransom. Jesus Christ paid our sin debt by his own blood and we are counted as righteous as God IN CHRIST.