I can understand why Refuge etc are unconvinced by it, and can't see it helping that many people tbh for the reasons given above (e.g. people not checking because they don't consider it a risk, "he's such a nice guy" etc).
Thinking about it a bit more, I reckon there are only 2 main scenarios where it could be helpful.
One is when someone has been warned about a new partner (e.g. by a previous ex, or family/friends) but is reluctant to believe it. IF there was a previous conviction, the ex would be able to challenge them to do the check to prove they were telling the truth; and in this situation I think (some) people might actually do it.
The other is after the new partner has started to become violent or just threatening. At this stage I think some women would ask themselves "hang on, does he have form for this?", and might be more likely to leave before worse happened if they found he already had previous convictions.
On the other hand, I agree that a lack of convictions could easily lead to a false sense of security if someone has been violent but not charged.
Also, if it's true about having to get the person's approval before running the check, I think this could actually be very dangerous. I could imagine some of the men I've read about on here wanting to "punish" their partners for daring to distrust them and "check up" on them 
(I was going to write "very dangerous in cases where the person is violent but doesn't yet have a record"... but then I thought, well, it could be equally dangerous if he does already have a record, but thinks he can get away with it...)