That thinking is time honoured and never found much traction. For example, pretty much nobody knows how to grow their own food, make their own clothes, carve their own furniture or even drive a manual car. Hordes of tourists circle the globe bringing disrepute to all sorts of time honoured monuments of history's greatest. Skills and challenges which aren't needed get forgotten and are generally not missed.
None of those things you mention have been forgotten. Many people do all of those things not because they have to but because it is incredibly rewarding to learn and grow these skills. Convenience doesn't bring happiness. People will actively seek out challenges even when they seemingly have none.
The trouble with things like climbing is there are only so many mountains to go around. We already can't walk in many places because of cars. I don't look forward to the day that similar vehicles can go up mountains. The existence of mass-produced clothing doesn't affect your ability to do your own knitting, though.
I'm replying almost solely to observe that inconvenience also doesn't bring happiness. Happiness is achieved precisely by feeling happy in the setting that you find yourself in. People can train themselves to only feel happy when inconvenienced but that is doing a major disservice to themselves and those immediately around them. But that is something of a tangent and so I have a cover story for why I'm typing!
> The trouble with things like climbing is there are only so many mountains to go around.
This is taking the metaphor far too far. Nobody is literally taking mountains away from people.
I think the majority of people still know how to grow their own food. We only passed 50% of the population living in cities a few years ago. https://data.worldbank.org/indicator/SP.RUR.TOTL.ZS says 43% of the population is still rural, and I'm guessing about 80% of that 43% (32%) knows how to grow their own food. So do all the people who have moved from the country to the city over the last 40 years.
There's a big gap between "pretty much nobody" and the reality, which is somewhere between one third and two thirds of everybody. You might want to reflect on exactly how your perception diverged so radically from reality.
Do people in cities suffer from not being able to grow their own food and make their own clothes? I don't know for sure, but official statistics claim that, even today, they commit suicide at much higher rates despite having much less material scarcity. Robinsonades have been a popular genre of fiction for centuries, suggesting that people long for that kind of autonomy. Today, we also have zombie apocalypse fiction, RPGs, and preppers.
From another angle, sports consist entirely of skills and challenges which aren't needed and never have been, suggesting that they don't get forgotten. Hobbies also consist of skills and challenges which aren't needed.
Um, no, that quotation means literally the opposite: it is about the climbing, it's not about getting to the top of any particular mountain. What Hillary was getting at is we get satisfaction from learning, training, overcoming difficulties and limitations, and ultimately pushing ourselves to our limits. His limit was Everest, your limit might be Snowdon, but it's climbing it that matters, not just taking the train to the top and taking a selfie.
He's saying that you can substitute any activity that combines danger, skill, and willpower for the mountain. It's literally not about the mountain, it's about how far you push yourself to reach a goal.