Yes.
Ultimately, the SECDEF cares about the POTUS's opinion on the matter, and the service chiefs are making policy from that guidance. From the SECDEF, we currently have an active stop-movement order and a mask mandate for military installations; neither of these include exceptions for vaccinated individuals. At the 4-star level, we have a laundry list of restrictions while in HPCON C - again, we don't have any exceptions for vaccinated individuals.
So we can all have a laugh when CinC wears a mask in a video conference, but the sobering reality is that it means restrictions on the military are not going away anytime in the near future.
Maybe not, which might be why no formal policy to relax liberty restrictions for vaccinated personnel has been promulgated, and everyone is still 'welded to the pier' when in port while in HPCON C regardless of vaccination status.
However, what can be done is COs can cite that there is an active stop-movement order in effect by the SECDEF / CNO that prohibits out of area leave or travel as the default, and if you want to take out of local area leave then one of the risk mitigations to authorize that leave would be vaccination because losing an extra 14 days to ROM is not tenable.