Schools lost a lot of classroom helpers fifteen years ago, because hardworking respectable families didn't want the school to know that twenty years previously they had a conviction for shop lifting or affray. DBS needs to be proportionate to risk, and needs to have some evidence that it reduces that risk.
Schools do need to DBS check adults who work with children.
The CRB system was formed specifically because of the most serious of crimes against two girls Holly Wells and Jessicca Chapman by the caretaker of Soham Village College, Ian Huntley.
There are a lot of myths and misconceptions about CRB/DBS (some can be seen on this thread)
The key issue is that there should be better understanding about Safeguarding, why DBS is needed and how the information will be used and held.
Those responsible for organising the DBS as they work with children and vulnerable adults will by definition understand and operate within professional confidentiality.
I've worked in environments where CRB/DBS was required. Yes, some volunteers had concerns and these were able to be dealt with appropriately and professionally.
If someone doesn't want to have a DBS check for whatever reason then no they shouldn't be working in the portential risk situations which require one.
The priority will always be the Safeguarding of the vulnerable.