Its interesting, because we have had a similar situation and a really good outcome. DS is at an independent selective/academic London day school. His personal organisation has always been a bit iffy - not shocking, just not great. It became clear that he was not, dunno, "marshalling" his work properly. He did not do well in his Y8 end of year exams (he failed a good few of them and was clearly floundering), partly as you say because he just was not effective in working out how to learn properly.
DH and I realised that some sort of academic support or intervention was required. We emailed the person listed as head of academic support on the school website to ask if any support was available. So far, so similar to you OP.
She emailed back within the day, and then spent 30 mins or so on the phone with DH the next day, by that stage having looked at DS's profile/results and also spoken to a couple of his teachers in subjects where he had failed end of year exams to try and understand the problem. Interestingly independently of this, his HOY also contacted us to say it may be needed (she was doing his report and it was quite obvious looking at the position overall that there was something out of kilter).
The head of academic support also supports learning programmes for academic scholars, and does learning plans with people who miss chunks of schooling, so seeing her is pretty neutral, which was great. DS was not in the slightest embarrassed about needing to see her - there was no stigma that I could see, she was just an available resource.
He had fortnightly sessions for a year, and is pretty much on track now. She has said she will see him intermittently, and I believe will be laying on additional "structuring your work" classes, particularly around exam time, for "invited guests" in addition to the usual ones the school lay on. But it was all very positive from our perspective.
So I think yr school is being a bit rubbish, tbh.