Yes I would. Where there is a financial relationship between the certifying authority and the applicant there is a much higher likelihood of bribery than there is via a government department.
Would you prefer a privatised police force ? Not that bribery doesn't occur in the police but if it was privatised there would be a lot less scrutiny.
How would the consumer know if the Authority does no checking ?
The old RSPCA certification for eggs got knocked on the head, didn't it, when people found out they weren't checking properly...?
Plenty of people will cheerfully blow the whistle on something like that, and the small cost / small (but a bit bigger) profit nature of that sort of thing would, I'd think, keep people reasonably honest, just because having your brand associated with dodginess is probably just not worth it.
Personally, I'm with you - any of that stuff should involve a fee to a government certification agency, rather than a company, with nice open books and auditing.
But it's a pretty damn minor issue, I would have thought, when in THIS case we have an Australian company being hit over the head with what is, in effect, bigotry.