Was the Kwisatz Haderach necessarily able to see the future?

Yes, the Kwisatz Haderach was necessarily capable of prescience. This was one of the key goals of the breeding program, to generate a human with the capacity to readily access the future prediction available to the Guild (and to a lesser extent to the Bene Gesserit themselves) without the attendant problems that each group faced; with the Guild, the loss of their humanity and tunnel vision on the problem of Spice production and for the Bene Gesserit, the inability to access the 'male' side of their psyche, and therefore the incapacity to use their prescience to dominate the known universe.

Paul said: “There is in each of us an ancient force that takes and an ancient force that gives. A man finds little difficulty facing that place within himself where the taking force dwells, but it’s almost impossible for him to see into the giving force without changing into something other than man. For a woman, the situation is reversed.”

_^Dune

Jessica's own agents prepared this summary of the Bene Gesserit's breeding program, presumably by interviewing Paul.

The Bene Gesserit program had as its target the breeding of a person they labeled “Kwisatz Haderach,” a term signifying “one who can be many places at once.” In simpler terms, what they sought was a human with mental powers permitting him to understand and use higher order dimensions.

They were breeding for a super-Mentat, a human computer with some of the prescient abilities found in Guild navigators.

_{^{Dune - Appendix III. Report on Bene Gesserit Motives and Purposes}}