There are several moments in that film that for the attentive viewer ought to have raised alarm bells even at the time let alone in hindsight. It’s pretty clear Theroux was briefed or had heard of the rumours and as you say he and his director did a good job threading through the potential legal minefields.
There is an interesting bit where, at a ship opening or something of the like, John Prescott appears to blank Savile. Savile walks towards him expecting the usual “hail Johnny well met” type of repartee but Prescott pretends not to see him and immediately turns to his right and engages a middle aged woman standing nearby in loud conversation. Though originally from Wales, Prescott spent 40 years as MP for Hull, first being elected in 2010, so it would not be at all surprising if he had heard rumours from way back when. Actually it would be extremely surprising if he hadn’t, particularly if he is remotely close to the champion of the working class communities type that he presented himself as.
I can remember browsing a “tribute to Jimmy” thread on a Leeds United Football Club fan forum around the time of Savile’s death and it was fairly clear there was a divide between the old Leeds hands who were well aware of the rumours and the younger types, or those not from the area who just happened to be supporters of the football club, who hadn’t an iota Savile was anything other than an eccentric old gent who used to be a big DJ back in the day and that did a lot of stuff for charideee.