“I learned a great lesson from Chappaquiddick … don’t drive over narrow bridges when you’re pissed out of your mind.” This gag from the BBC’s 1970s sketch show Not the Nine O’Clock News, with Griff Rhys Jones playing a sombre Ted Kennedy, perfectly encapsulates his cynicism and self-pity. It seems to me more apposite than this lenient movie about Chappaquiddick, which now belatedly appears in the UK, starring Jason Clarke as Kennedy, scripted by first-time feature writers Taylor Allen and Andrew Logan, and directed by John Curran.

While driving back in darkness from a party at Chappaquiddick Island in Massachusetts in July 1969, Kennedy’s car went off a narrow bridge into the water. Kennedy swam clear, but his passenger drowned: Mary Jo Kopechne, a woman who had worked as a researcher for his late brother Bobby. Having failed to report the incident for eight hours and toyed with the idea of claiming Mary Jo was driving, Kennedy finally admitted to leaving the scene of an accident, accepted a suspended two-year sentence and gave a televised address claiming he hadn’t been drinking and there was no impropriety with Kopechne. The US had no great appetite for spoiling the moon-landing euphoria with a political scandal, or spoiling the tragic Kennedy mystique . So the media and political classes suppressed their distaste and embarrassment, allowing Kennedy to plough on with his long but undistinguished senatorial career and a failed bid for the Democratic presidential nomination in 1980.

Clarke plausibly plays an entitled and defensive Kennedy, his flushed face set in a rictus of fear and unrepentant resentment; Kate Mara is Kopechne, and Bruce Dern is the macabre figure of Joe Kennedy Sr, incapacitated by a stroke and seething with mute rage and contempt for his deadbeat son. It’s a movie that in important ways is on Ted Kennedy’s side. It does not show him drinking heavily, or drinking much at all at the party, and does not show him kissing or having sex with the similarly non-drinking Kopechne; just being very close and intimate about his fears and plans. And when the car starts going out of control, it is Mary Jo’s panicky gesticulation that obscures his view of the road.

But the film shows how the family courtiers and spin doctors rallied round: from big cheeses like the former secretary of defense Robert McNamara (Clancy Brown) to Ted’s beta-male cousin and fixer Joe Gargan (Ed Helms). It persuasively suggests how Kennedy did indeed retreat into a stunned state of denial, and that the catastrophe of Chappaquiddick jolted him into something that looked almost like political seriousness. Namely, his traumatised horror of prosecution, a mannerism that he carried over into the long anticlimax of his political career.

Yet The Senator is rather prim and prissy in its reluctance to acknowledge what is under its nose and what has been under American history’s nose for more than 50 years: the possibility that Ted Kennedy was just very drunk and that he had had or was eager to have extramarital sex. And the film could have done more to dramatise Kopechne’s life and show that she was something more than a footnote. Some interesting moments here, but the film is one long pulled punch.

• The Senator is released on 19 March on Amazon Prime Video.