Have tried to read most of the comments on here but may have missed a few - it is really interesting and a huge issue. I have experience of being a University Faculty Officer, a residential tutor in a Hall of Residence and a Student Counsellor, ( all RG Universities and the OU) and I do think this question needs careful thinking about by Unis in general- there are so many aspects to consider, confidentiality being perhaps the major one - although the issue of parents paying fees/living costs etc is now really raising its head, as many see a University education as something to be 'bought' - as a value for money kind of investment. I guess my perspective is from the students themselves, - the actual hands on living issues faced by students daily, including personal health, finance issues, accommodation issues, coping with being independent supposedly etc, and seeing how this affects academic issues. There is the question of whether an 18 year old's transition to being an adult is a level playing field if that makes sense, some are ready, mature and capable, some are immature, out of their depth, and struggling - just because they have physically turned 18 and have done well in exams is not necessarily adequate preparation for then living independently and managing things.
I have been involved with students on medical courses, many successful, but some of whom have tread water from day one, and the only 'legal' communication from the university would be down to exam failure/non attendance etc,. There needs to be proper pastoral/tutor time with individuals for appraisal and review and this often does not happen enough. It is impossible for many 'strugglers' to discuss issues with tutors, or indeed parents, and although Universities do offer counselling and other types of support, those who need it most many often be the last to take these services up.
As a residential tutor, I was 'available' to students at all hours and had regular open door evenings for them to come and discuss whatever, in a confidential way, sometimes it could be sorted easily, often more complex issues had to be dealt with by other services.
Things have changed enormously in the past 20 years or so, and our 'old' experiences are long gone really - the Uni/student/parent/ interface is a really complex issue which these organisations must tackle.