Before she took over Woman’s Hour in 1987, Jenni Murray was a presenter on the Today programme. She had joined the BBC in Bristol in 1973, and became a TV reporter and presenter for South Today, so arrived with solid news credentials. But Today in the 80s was inveterately sexist – the guys took the politics, the women mopped up the rest – that the format was just too small for her.

Woman’s Hour, on the other hand, was absolutely reshaped in her image: there was no preconception of tone, and nothing was too serious or too light for it. Murray, who has died at the age of 75, could tear a strip off a politician, talk about hydrangeas, then campaign against domestic abuse, all within a few minutes. She was instinctively open and generous about her personal experience, but never solipsistic – an incredibly fine balance.

People always used words like “mellifluous” and “rounded” about her voice, which makes it sound like an elocution triumph – RP scores again – but that was a misunderstanding. The main barrier to entry when she joined the BBC was not sounding posh enough. You had to speak that way, whether you were posh or not (and she was not – her dad was a civil engineer, her mum a civil servant).

Her warmth filled the kitchen every morning, after the show moved from the afternoon in the early 90s. It was married to a natural beadiness and wit. You knew you were not going to be bored. Murray would reminisce about her big political interviews, of which even the so-called failures were remarkably listenable.

When she spoke to Margaret Thatcher after her departure from office, she asked her if the sexism in public life had ever bothered her, rattling off some of the more hair-raising remarks: Alan Clark calling her ankles pretty, or François Mitterrand saying she had the eyes of Caligula and the lips of Marilyn Monroe. “She looked at me and didn’t speak,” Murray said later, and it dawned on her that the former PM might not have seen those comments, and “when I put those things to her, she was genuinely shocked”.

Her unabashed probing of Hillary Clinton – she really went there about Bill’s infidelity – yielded probably the most human sounding interview Hillary had given. Marriage was about friendship, she said, not necessarily about sex. It was not a groundbreaking statement, but it felt truthful to that couple. Murray was rarely starstruck, though years later still remembered her terror at interviewing an 81-year-old Bette Davis.

She had an instinctive female solidarity, and a keen eye for the young feminists coming up behind her. One show in 2013 – with Caroline Criado-Perez, then a novice campaigner for women’s placement on banknotes, Laura Bates, founder of the Everyday Sexism project and Allegra McEvedy, a chef pondering the quintessentially feminist question of what foods are best on a hangover – was a pretty typical day for Murray, yet it distilled her acuity and determination to talk about what mattered, with people who made a difference.

Murray was diagnosed with breast cancer in 2006. She hated talking about it, finding it boring, and yet she had to talk about it. “I couldn’t just disappear to have a mastectomy,” she said in a Radio Times interview. “You develop a closeness to your audience.” And with Murray, that was a two-way street.