Even the strong 'in a perfect theoretical world' version only says;

> The theorem states that if trade in an externality is possible and there are sufficiently low transaction costs, bargaining will lead to a Pareto efficient outcome regardless of the initial allocation of property.'

Key phrase there being 'Pareto efficient' which basically means 'without addressing the initial starting point inequality'.

I'm also not sure what your last claim is about, it sounds like a garbled version of 'you need inequality to progress' which is a common trope, but not actually backed by any logic as far as I can tell, or even fleshed out enough to escape from the 'not even wrong' category.