This weekend, while the Duke and Duchess of Sussex were arguing the toss with Palace officials over the terms and conditions of their forthcoming visit to Britain (will they or won’t they bring the kids, which palace will be at their disposal, how much security will be laid on and so on), the Princess of Wales – an actual working royal – was quietly, without fanfare or a large film crew, climbing three mountains. Actual mountains.

Arguably she has already climbed several metaphorical peaks over the past two years following her cancer diagnosis, not to mention dealing with a load of nonsense from her sister-in-law, but these were real ones: Ben Nevis, Scafell Pike and Yr Wyddfa, the three highest in Scotland, England and Wales.

It was part of the Three Peaks Challenge, in which participants have to scale the trio within a 24-hour window. This involves walking more than 23 miles and hiking up over 10,000ft, plus, of course, driving 462 miles between the destinations. Along the way she bumped into a wheelchair-bound 11-year-old named Ted Haslam – paralysed at the age of just three by an aggressive spinal tumour – whose father was carrying him up Ben Nevis.

Otherwise, she was pretty much on her own, save for other climbers and a couple of cameramen.

No Press briefing, no paparazzi, no crowds of well-wishers urging her on. Just a quiet, determined, low-key act of personal pilgrimage.

It was all in aid of the Royal Marsden, where the Princess herself was treated for cancer, and in particular to fund holistic ‘whole person’ care, which doesn’t just focus on clinical treatment but also on helping the patient navigate the emotional and physical challenges involved. The whole enterprise was what you might call ‘peak Kate’: a masterclass in how to do something worthwhile without turning it into a tawdry media circus or making it all about herself.

Even her subsequent statement on the Wales family’s official Instagram account was carefully calibrated to draw attention away from the Princess, and on to the work of the hospital and charity.

Only once in the statement did the Princess obliquely refer to her own health: ‘I know this personally, and that the journey through and beyond treatment requires more than medicine alone.’

This is exactly the kind of thing we expect from members of the Royal Family, but which in recent years has sadly been in short supply. Here is someone using her platform to genuinely help others, working hard to earn the respect of the British people, and honouring with her actions the titles and privilege her royal status affords her.

Here is someone, as the young people say, who understands the assignment. That is why words like ‘grateful’ and ‘give something back’ ring true with the Princess of Wales, but they don’t with others.

For example, compare this authentic, low-key approach with that of the Duchess of Sussex, who operates at exactly the opposite end of the spectrum.

She spent this weekend teasing a new product on her As Ever website (lavender honey, since you ask), alongside the usual carefully curated shots of her posing pensively in her kitchen, caressing her own neck with a fond, faraway look in her eye.

Honestly, if she were an ice lolly she’d lick herself.

Everything she says or does, whether it be accompanying Prince Harry on some pseudo-royal tour or visiting a women’s shelter, is not about the people she professes to care for, but really about her. Try as she might, she just can’t hide it.

Remember that seven-minute-long charity speech she gave in which she mentioned herself no fewer than 54 times? Or the ‘fairy dust’ comment on a podcast (‘If I can come in with a little sprinkle of fairy dust just by wearing a bracelet... that’s a huge honour to be able to have that kind of impact’)?

Even when she is trying to be magnanimous, her narcissism shines through.

In a way, it says a lot about society today that the Princess of Wales’s low-key challenge catches us all by surprise.

We live in a world where increasingly a person’s value is not judged by who they are or what they do, but by how well they can project themselves on social media and how much attention they can garner from total strangers.

It leaves us feeling strangely disconnected, detached. It’s like being at a party surrounded by people, none of whom you know. Plenty of contact, no actual connection.

The Princess of Wales’s approach is different. She doesn’t chase popularity, or cast herself as a victim to make people like her (even though, having been so very ill, she could).

She just knows instinctively how to do the right thing – as the incident with the little boy on Ben Nevis shows – and then everything else follows. That’s the kind of personality you just can’t fake, that no amount of artfully staged camera angles can reproduce.

In a funny way it reminds me of the approach the late Queen Elizabeth took, quietly ploughing her furrow based on a set of universal moral principles.

People didn’t love her because she wore the crown; they loved her because she had the right values and understood what being royal really meant.

Perhaps in the Princess of Wales she has finally found a worthy successor.