I don't think it's a good idea to turn off the background lines in this case. I also think the dimension is fine as originally presented. The lines do not obscure anything and they are an accurate representation of what the part looks like from that view. Turning the lines off won't clarify anything.
I have seen cases where turning tangent lines either on or off makes a view more clear but I don't think I've ever seen a case where removing lines representing actual features did more good than it caused confusion. I always expect any view to accurately represent what the part will look like. I agree that excessive dimensions in one view can make a cluttered print but that's not the case here.
If you look at the parent view in the first print you provided, there is no dimension for the OD, ID, or counterbore. Will turning off the lines to those features in your parent view make the drawing more clear since those lines aren't "adding information"? To me, it would be a much more confusing print. I just don't think it's a good idea to turn off lines depicting features. I'm sure there are some cases where it makes perfect sense but this isn't one of them. I think that hiding feature lines should be more the exception instead of the rule.
Powerhound, GDTP T-0419
Engineering Technician
Inventor 2010
Mastercam X5
Smartcam 11.1
SSG, U.S. Army
Taji, Iraq OIF II