My children are bright so I am not worried about them, but I suspect those who aren't brilliant but quite capable of doing better lose out. I'm all for care and concern and the school is brilliant with special needs, but the form tutors and year heads seem wholly concerned with personal problems and behaviour.
I don't know where in the structure overall progress gets watched.
I suspect there are quite a few schools like yours, despite government pressure for results. Pastoral care became a big industry in the early days of comprehensives, and it is vital that young people's welfare and happiness should be of prime concern if they are to achieve their best. But keeping ends and means in the right balance is important too.
Imbalance can often be detected just by looking at the management structure and the job descriptions associated with it. It could be that your school needs a serious review of that structure, but this is a very difficult thing for governors to initiate unless they, or at least a governor or two among them, feel easy enough with the head to sow the seeds. Also this is a long term solution and not always painless.
I agree that form tutors and year heads are the front-linrs to monitor overall progress and liaise with subject teachers to that end. It's probably also a good idea if there is one senior member of staff who, with year heads, analyses all the performance data to pin-point under-achievement. These duties should be absolutely clear in job descriptions and there should be enough time built in for them.
Middle-management posts especially may need more clarity, but realistically it isn't easy for governors to involve themselves except when staff changes take place. I don't know how much your governing body is involved in teacher appointments and promotions, but these are times when the emphasis can be changed.
Also when governors join in a review of exam and test results - and they should - there may be justification for bringing up the question of overall progress monitoring as a key pastoral responsibility. Your target-setting role can also be fruitful, and you may need to be a bit firmer when you think teachers are being too cautious, using performance and data reports and other comparative studies for evidence that you could do better. Many schools are now also using individual targets for students.
In a school with first-class management all this would seem elementary, but from talking to many governors I suspect that there are some schools where pastoral posts are not sharply enough directed towards academic progress.
It's a delicate area for governors: discuss what I have said informally with colleagues as a first step.