Fox News personality Pete Hegseth, a conservative voice on veterans’ policy, has emerged as a leading candidate to replace embattled Veterans Affairs Secretary David Shulkin, who has fallen from favor with the Trump administration, according to people with knowledge of the matter. Hegseth, 37, is co-host of “Fox & Friends Weekend,” a platform he has used to push his vision of a health-care system with a drastically smaller government footprint and a larger share of private care.