Your petition isn't going to get anywhere because what you want exists now.
Every stage of the house build is inspected by either local authority building control or a private organisation accredited to do it such as the NHBC. In addition there are warranty stage inspections which are more thorough.
This of course depends on whether they are actually carried out as required by law the Building Regulations, and the competence and diligence of the individual inspector.
As for redress, there is now a New Homes Ombudsman, something I spent 8 long years campaigning for.
It is not currently statutory, and is questionably independent, but it now exists, thanks in no small part to my own efforts.
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