WCollins
Another factor is the Required Navigational Performance (RNP) of the airspace. Certainly in BRNAV airspace in the UK at the moment it is RNP5 which means you can wander outside of track by up to 5 miles for 5% of the flight quite legitimately.
Whilst practical experience shows that this 'allowed' wandering rarely happens, basing 'radar' separation of 3 or 5 NM between aircraft each of which can in theory be anywhere on a 10NM wide swathe in congested skies is not a wise idea. Hence we cover our ar*es and yours by using assigned headings.
I'm not even sure that the TMAs such as London below the BRNAV airspace even have an RNP accuracy assigned, thus the margins for error could be even greater depending on what the pilot is using to navigate
Again unlikely but the legislation and procedures always cover the worst case.