At my previous employer, the largest 737 operator in Europe, you could only accept a visual if you had briefed it and double briefed it, and you had to be lined up with the runway at 4nm out. So in theory, if you hadn't briefed it, you could not accept..
As you say, 'in theory'. If they turned off the ILS when you were at 10d on a clear day, would you not continue?
With an ILS or VNAV/LNAV profile briefed, a 'visual' simply becomes a shortened approach with either a self-positioned segment or radar vector to the 3 degree path. The higher minima (if that's what the company require) can be set in a second. In both cases, I expect a 4 mile final would be most people's preference?