Strictly speaking a Praefectus of the Cohortes Praetoriae did not have a role encompassing the Empire, he was commander of a bodyguard in Rome, though individuals were used as spies and a detachment of Praetorians would usually follow the Imperator when he journeyed - but then many of these leaders never left Rome anyway.
The Praetorians are often blamed for unstable rule, not without good reason, but in most cases it wasn't the soldiers themselves, but rather the Prefect who had close contact with elite men in Rome who was instrumental in court intrigue. The face-off of Praetorians against the Senate that got Claudius in power, or the auction that did the same for Didius Julianus are obvious exceptions.
Please note that the wiki article is incorrect. The Praetorians did not exist in the Late Republic, it was Augustus who formed them from all the bodyguard units left over from the civil wars, and the Urban cohorts were made from Praetorians with their own Prefect who doesn't figure in the historical record much, I can only think of the takeover immediately after the death of Caligula which the Praetorians overturned.