I see your point, and it being a correct one, about the Elders Quorum being the only place that members of the HP quorum can report their attendance. However, under the additional FAQs for ministering (question # 8. link to page below) members of the HP quorum are not generally assigned as ministering brothers with specific families, though ministering brothers and sister can be assigned the HP’s family:
“Under the direction of their bishop, the presiding high priest in the ward, the members of the high priests quorum and their families have ministering brothers and sisters assigned to them by their elders quorum and Relief Society.
Because stake presidencies and bishoprics are responsible for all members of the stake or ward, these brethren are not generally assigned as ministering brothers to specific individuals or families. High councilors and functioning patriarchs may be assigned, based on local circumstances, as determined by the stake president. If they are assigned, it would be by their elders quorum president under the direction of the bishop of their wards...”
Our Stake President specifically asked us recently to make sure the EQ Presidency reassigns any families assigned to us.
As for the belief that members a different quorum are simulateously members of their ward EQ, that doesn’t jive with what Elder Christofferson said as quoted in this thread, at least to me.
https://www.lds.org/mycalling/ministeri ... q?lang=eng