“I’m a painfully truthful person often to the point of self-immolation or self-destruction,” said Julie Burchill appearing on Desert Island Discs. Burchill was bound to draw an audience, and only a few weeks after the BBC programme hosted its dream guest Aung San Suu Kyi.

We were certainly in for some self-immolation. Burchill’s mantra was “I don’t care”, from talk of abandoning her son right to the last incomplete sentence.

The trouble is, perfecting “I don’t care” is a contradiction in terms. And when a media personality is so self-consciously truthful as well as painfully so, we begin to wonder. She boasts of the distance between her “good twin” – represented by her spoken voice – and her “evil twin” – the aggressive voice of her columns. Could the void between her broadcasting “truth” and that of reality be just as wide?

But sincerity or pretence, Burchill’s appearance on Radio 4’s flagship magazine programme was compulsive listening. For those of us who hold her views – overt transphobia among them – in contempt, perhaps even more so. With construction and self so entangled in a life she would admit has been particularly messy, the bones of her insecurities were laid bare.

Constructed or not, the “truthful” Burchill was ever-present in her choice of tracks. No less than three were in line with her extreme Zionism, which has recently led her to announce that her “only criticism of Israel is that it could stand to be quite a bit bigger.”

Needless to say, the one disc she opted to save was the Israeli national anthem.

In a less worryingly-fanatical touch, she justified her choice of Spike Jones’s classic Cocktails for Two: “I don’t just like to have a glass of wine and relax, I don’t just like to sip at a single-malt whiskey, I like to get drunk, and this is a very funny song about this.”

Too often on Desert Island Discs, such opinionated guests are encouraged to expound their philosophies and ideological viewpoints, and yet rarely face challenges on anything except the most personal. Even here, the tone remains cosy: and in the main, that is perhaps the programme’s success.

Taking Burchill up on her transphobia would, alas, be too heavy under this interpretation of the rules. A layer of severe discomfort joined this episode’s unique captivating and bizarre nature. It’s great for guests to exfoliate their emotions and weaknesses, but even the genuinely honest cannot do so objectively. And our broadcasters should never allow prejudice to be brushed under the carpet.

It was also a missed opportunity. A challenge to prejudice and fanaticism would not have raised the tension in the studio: it was at peak heights anyway. Denying she cared for the feelings of those who cared about her, Burchill snapped: “I can tell why you’d think that, because you’re a nice, civilised person, but I’m not like you.”

To which presenter Kirsty Young retorted: “You don’t know me.”

The intense personality in the hot seat had made a striking programme to begin with. Her “sweet and childish” (Young’s words) voice was unusual for Radio 4 – and after we acclimatised, quite disarming. Her candid talk of interminable illegal highs and vulnerabilities added an unusual touch. But the highlight was certainly an intensity of exchange one rarely finds on the Today programme, let alone Desert Island Discs.

Burchill opened the programme with a claim that desert island life would be “ideal” for a misanthropist like herself – although she soon talked of her husband and her Hebrew classes as her week’s highlights. “I would lie on the beach, reading the book, getting drunk on this cocktail, listening to the Israeli national anthem,” she concluded. There are plenty who would afford her this warped sense of bliss so long as the rescue boat never came – and understandably so.

Yet Radio 4 would have been a poorer place for it. Do we really need to like or agree with a programme’s guest to find it totally enthralling? Perhaps all comparable guests were marooned years ago.

Listen to Burchill’s Desert Island Discs appearance here