It is the base commander that has prohibited concealed carry on base, not DoD policy.
DOD DIRECTIVE 5210.56 ARMING AND THE USE OF FORCE
SECTION 4: POSSESSION OF PRIVATELY OWNED FIREARMS BY DOD
PERSONNEL NOT RELATED TO THE PERFORMANCE OF OFFICIAL DUTIES.
4.1. GENERAL. This section provides guidance for determining the eligibility of DoD personnel to carry privately owned firearms on DoD property for personal protection when it is not related to the performance of official duties. This section also provides requirements for individual training, transport, safeguarding, and storage of privately owned firearms that the arming authority must consider when determining whether to permit an individual to carry a privately owned firearm on DoD property.
4.2. ARMING AUTHORITIES CHARGED WITH PERMITTING PERSONS TO CARRY PRIVATELY OWNED FIREARMS. The arming authorities charged with determining whether to permit DoD personnel to carry privately owned firearms on DoD property:
a. May grant permission to DoD personnel requesting to carry a privately owned firearm (concealed or open carry) on DoD property for a personal protection purpose not related to performance of an official duty or status. Permissions granted under this section do not apply to carrying a firearm within federal buildings unless the arming authority specifically determines, after consultation with servicing legal counsel and in accordance with applicable DoD policy, that an appropriate exception under Section 930(d) of Title 18, U.S.C. applies.
The current directive you cited requires 0-6 or GS15 approval, on a case by case basis. As the article notes, it is very strict (i.e. difficult to obtain such permission)
For example - take a Corporal (E-4) in an infantry unit in the Marines. Suppose he is a team leader by billet. In order to get such authorization, he would have to talk to his squad leader (E-5/Sgt), who would then ask the Plt Sgt (E-6/SSgt), who would then talk to the Plt Cmdr (O-1 or O-2/2nd or 1st Lt).
The Lt would then ask the Company Cmdr (O-3/Cpt). The Co. Comdr would then ask the Battalion Commander (O-5/LtCol), who would then go to the Regimental Commander (O-6/Col) who, by the Directive is the first person in the Corporal's chain of command with the authority to grant authorization, and that's if the Base Commander concurs that an O-6 can make that authorization.
Along the way, the Company Gunny (E-7) and/or the Company 1stSgt (E-8), Battalion SgtMaj (E-9) and likely Regimental SgtMaj would want to speak with the Corporal.
In other words, a snowflake has a better chance surviving in hell than the average service member's chance of getting the approval to carry personal arms on base.
Further proof - When I was stationed at Lejeune the Onslow County Sheriff at the time required a "Command Letter" granting the authorization for a CCW permit. NC law does not require military members to furnish a "Command Letter." The Base did not require such, nor did the tenant commands. My command asked, "What the hell are you talking about?"
The Sheriff wouldn't issue the permit without one.
It finally came to a head and the Base Commander and MEF Commander caved - and it became required (on the military side). The surrounding counties followed suit and their Sheriff's began requiring the letters too.
When I was subsequently stationed at Quantico, the Base Directives made it more difficult to transport/possess weapons on base as well.
The Directive you cite does NOTHING to enable the protection of U.S. service members on base. All it says is that the commanders may.
The policy sets the command level required to approve high enough that the average joe will never get authorization.