Actually, the timeline does support the theory. The historical nucleus of the Metropolitan Police is the force organized by Henry Fielding, in the 1750s, when the Wild scandal was still on everyone's minds. That provided a template, and the growth of London in the late 18th/early 19th century meant that local forces were unable to keep up. In 1830 or thereabouts the local constabularies were amalgamated into the Met.

As for the corruption bit, with private security forces we know the weakness from the historical example already. With public police, public oversight puts a limit on corruption, at least in theory. Certainly London saw an improvement with the government providing police service.