I expect the answer you will most gravitate to is hinted at in your choice of words. "Forge" certainly denotes something made to be strong, but it also evokes an ironically antithetical image (since to forge something is also to create a fake).

As drawn from a metallurgic context, a forge is to shape and strengthen a metal through repeated heating and beating. It is forceful, exhausting work that creates objects of immense strength and ingenuity. Some might guffaw at the idea that this same sense applies to families, but I would argue that they are taking for granted the benefits of experiencing adversity as a team as well as the benefits from strong, purposeful leadership. (Disclaimer: do not heat or beat your families.)

I think a father can bring this quality to the family by way of introducing the family to experiences that forge their bonds and other strengths. Family vacations, adventures, chores, charitable acts, moments of humility and forgiveness--all of these situations and more can test a family and have them coming out the end of it stronger. The father's role can be knowing what tests and experiences are appropriate for his family and to be their protector through it. I think it would also be his challenge to know that he is not separate from these tests.

Another verb you could have used is "grow." There are plenty of phenomena in nature that show how growth can lead to remarkable strength. An oak tree comes to mind. This might be a word better suited for a mother (though a father is not completely exempt). Growing requires patience, nourishment, and sacrifice--these are also motherly traits that might contribute to a strong family.

I have more to say on the matter, especially more specific to your questions, but I leave this reflection on the question itself for now.