I found it curious when reading Kitchener's bio that he and his fairly large party were embarked on HMS Hampshire. Couldn't the Admiralty provide a more robust, larger vessel? I'm not trying to get into the many conspiracy theories about the death of Kitchener, just find the choice of vessel odd, not to mention departing in heavy weather.
In June 1916 the Grand Fleet wasnt exactly swimming in extra vessels to send on a diplomatic mission, an obsolete armored cruiser was about the most Jellicoe would have parted with under most circumstances.
Following Jutland the situation was only heightened as with so many ships shot to pieces and in need of yard time the need to keep every worthwhile vessel on strength was a matter of priority. 3 of Beatty's battlecruisers were sunk and 3 of his 5 newest battleships were shot to pieces in Barham, Malaya, and Warspite. And while the main body came through relatively intact Jellicoe was ever sensitive to attempts to call away parts of his strength and placing his margin over the High Seas Fleet in jeopardy.
So that basically rules out any of the capital ships especially in Summer 1916. While there were never enough light cruisers and those that could be pumped out were coveted by Jellicoe and Beatty to serve as scouts and destroyer flotilla leaders.
Meaning the armored cruisers were basically the only ships on hand to the Admiralty that could be sent on special missions. Pre-Dreadnought battleships might have been an option but most of them had been scattered to the winds, either still in the Mediterranean following Gallipoli or further afield as station ships with every modern warship possible called home. But while some were attached to the Nore Command at the time they were generally slower, and less comfortable rides with lower freeboard, so non ideal for the need at hand.
So back to the armored cruisers, ships with dubious military value essentially just on strength to fill space and help enforce the blockade. While often they had been designed with service on foreign stations in mind so they would have the modicum of comforts to make the voyage, and extra accommodations for an admiral and staff. This lack of a mission in battle was reinforced at Jutland when Admiral Arbuthnot got himself and 3 of his 4 armored cruisers blown to Hell when he took them between the battle lines.
And in general the RN was always hard up for ships to meet their needs, to the extent of even using auxiliaries and called up merchant vessels early war to conduct patrols to enforce the blockade in the North Sea. Part of this was simply the scale of the mission, but the hoarding of every modern vessel Jellicoe and Beatty could get certainly contributed as well.