Goolagong opens to the soulful strains of Ann Peebles proclaiming: “It’s your thing – do what you wanna do!” It feels a little on the nose as a way to soundtrack an inspirational sporting drama, as Australia’s Evonne Goolagong (played by Lila McGuire) steels herself for her first ever Wimbledon match. (For the uninitiated: not only was Goolagong the first Aboriginal player to compete in tennis’s most prestigious tournament, but she would go on to win the ladies’ singles title twice, in 1971 and 1980, plus a doubles win in 1974. She won seven grand slams in total and was – for a time – ranked world No 1.) This three-part drama from Australia’s ABC is sometimes saccharine, and the opening sequence of a teenage Evonne wandering starry-eyed through the corridors of the All England Club – portraits of former winners on the walls – feels heavy-handed. More difficult themes do come to the fore in time, but Goolagong is largely an unapologetic, flashback-heavy tribute to a sporting legend. It’s beautifully drawn, but do we really need to watch the primary school-aged Evonne (a cherubic Eloise Hart) hit a ball against a wall with a plank of wood this many times?!

Sadly, being a woman in sport – or maybe just a woman in the world – Goolagong would go on to apparently suffer financial abuse and sexual harassment at the hands of her coach, Vic Edwards. The contrast between those fluffier scenes and the unwanted advances of Marton Csokas’s slippery Edwards feels like a screeching handbrake turn. Not least because we see Edwards move Goolagong from her happy but impoverished Wiradjuri family in rural Barellan, New South Wales – with a population in the hundreds – into his family home in Sydney at 14, grooming her for sporting fame but also maybe just grooming her full stop. But – as uncomfortable as that segue is – it is her reality. “When it stops being fun, come home,” Evonne’s mother tells her, with more than a little foreshadowing on the part of the writers. Later, after family tragedy and chicanery on Edwards’s part, Evonne will echo those words, declaring that tennis is “not fun any more”, ruined by the selfishness of her mentor.

Elsewhere, the series does well to weave in the big issues that overshadowed the game in the 70s – and conversations on race, gender and pay equity – without feeling too much like a rehashing of Goolagong’s Wikipedia page. McGuire is brilliantly believable as the clueless upstart who isn’t unfeminist, but sticks her foot in her mouth by telling a journalist that she would play for free if she had to. (Naturally, Billie Jean King is deeply unimpressed, and Goolagong finds herself ostracised by her fellow women players.)

As syrupy as some scenes can be, they are anchored by the brilliance of Hart, McGuire and Rilee Clarke, who play Goolagong as a defiant, determined, quirky woman at different points of her life. The supporting cast is strong, too – in particular Luke Carroll as Goolagong’s father, Kenny, and Chenoa Deemal as her mother, Linda. And who can resist the burgeoning, trans-hemisphere romance between Evonne and English tennis journalist Roger Cawley (Felix Mallard), who would go on to become her husband? (Well, perhaps Vic Edwards could – allegedly, Edwards lied about not being invited to the wedding, then unilaterally announced Evonne’s retirement.) Even so, Goolagong can’t quite make up its mind tonally. The result is a drama that’s frequently charming, but frequently lightweight. Certainly, the crescendo of the final episode – and Goolagong’s return to the sport just months after the birth of her daughter, Kelly, in 1977 – drags on and on with a tension that feels forced. All before a miraculous recovery, a family reunion and that joyous second Wimbledon win. Hurrah! Cue more flashbacks …

Goolagong is an uneven thing, although clearly it’s a story that wholeheartedly deserved to make it to the screen. It ends with a slideshow of images of the real Evonne, which only confirms my sense that a documentary or docudrama would have been more compelling. We are told that she “seeks out a new generation of talented Indigenous children” through her tennis charity, “supporting them to dream, believe, learn and achieve”. She and Roger have now been married for 51 years. There’s a brief clip of her with McGuire, as they wave to a crowd of extras, that is rather moving. It’s not quite smashing, then, but it is lovely.

• Goolagong aired on BBC Four and is on iPlayer now.