Re: I would agree with only LEAs having full access
"The real-world details of British businesses are searchable via Companies House so why not their online details too?"
I think you will find that you have been able to redact that information for quite a while now. Many companies and their directors now simply use (legally) an accommodation address - typically their accounts or solicitors and not their own.
Nominet's proposals are rather less than Companies House already does.