The NZ Rugby 7's players struggle to make the Soup squads (perhaps because of the time commitments required by the 7's programme), but they are nothing outstandingly brilliant in the ITM Cup. Victor Vito was a 7's player initially. While he has made the AIG's, he has struggled to be considered first choice for the AIGs in the 15 aside version.
At some of the bigger schools, there is not much difference between a player in the 1sts and one in the 2nds or 3rds. The 3rd best 7's squad from Joeys would most likely be capable of performing with distinction at a State 7's, with little impact on the overall rugby programme at Joeys, even if the tournament was in the middle of the AAGPS season.
Having seen most of the Rep Schoolboy games this year, Barker (the bulk of the CAS rep team) and St Augustines (the bulk of the ISA rep team) would be able to stand up a very competitive Rugby 7's team. From AAGPS, Riverview and Newington would similarly be able to run out teams that be able to compete with the LLoydies, Scots, Sports Highs, and St Gregs.
Like the Waratah Cup/Shield competition for Schoolboy 15 aside bragging rights, the competition is poorer for the absence of certain teams/schools.
As said above, it is easy to criticise from this side of the keyboard in the absence of all the facts. The schools in question may have decided not to nominate themselves, or may not have known about this in advance when they finalised their sports plans for the year. There is a very tight turnaround between the Winter and Summer sports programmes at the AGPS, CAS, and ISA Schools associations.
The parent body associated with Barker, Auggies et al should perhaps be asking a question or two of their sports/rugby masters, and suggesting that they should perhaps consider nominating for next season's tournament.