If Southron Ambitions was an attempt at a Magna Carta and/or a permanent Great Council that puts more power into the hands of the Lords Paramount, why didn’t Jon Arryn implement the idea after Robert’s Rebellion?

A good question, and one of the better strikes against the theory. 

A couple potential answers:

1. Robert wasn’t on board, and with Rickard dead, Eddard in the North, and Hoster injured, there wasn’t enough of a coalition to push him. 

2. Jon Arryn changed his mind once he had the royal government in his hands, and the king was no longer a violent lunatic.