Here's the bottom line:
Military chaplains pray in two different types of circumstance. They pray when they are conducting religious services for people who share their beliefs, and they pray in circumstances when people who do not share their beliefs are required to be present. Nobody is questioning the right of chaplains to pray as they wish when they are conducting the services appropriate to their own denomination. However, when praying in a situation when people from other religious traditions are present, it is entirely inappropriate for a chaplain to deliver anything but a non-denominational prayer.
The executive director of an organization that represents some evangelical chaplains and is pushing for the executive order is quoted in the Washington Post article linked above:
But Baugham said evangelical chaplains must represent the church that endorses them for military duty, and "they are not authorized to give nonsectarian prayers."That position is extrordinarily odd, particularly coming from a group that endorses chaplains for military duty. According to this army website, a chaplain's endorsement from their faith group should certify that the candidate is, among other things, "Sensitive to religious pluralism and able to provide for the free exercise of religion by all military personnel, their family members and civilians who work for the Army."
Any chaplain that cannot be sensitive to the differing beliefs of others in their unit is simply unfit for the needs of the military, and should be discharged.