Ticket to watch Simone Biles at the Arena Bercy: €648 (£550). Ticket to watch the men’s 100m final at the Stade de France: €980. Ticket to watch some of the world’s greatest cyclists racing around one of the world’s great monuments in one of the world’s great cities: gratuit. No QR codes. No searches. No private security sentinels checking your bag and asking you to drink your water to make sure it is, in fact, water.

Which is not to say that your admission to the men’s cycling road race on the Butte Montmartre is going to be painless. For one thing, there are the hundreds of steps you will have to negotiate from Abbesses metro station, a physical trial that will have you reaching at frequent intervals for Jens Voigt’s favourite maxim (“shut up, legs”). Then there are the queues and the crowds, the countless extra minutes that Google Maps does not calculate. But – after a fashion – here you are, on the slopes of the Sacre-Coeur basilica, ready for a feast of pain and pleasure.

Related: Remco Evenepoel survives late puncture to take historic Olympic road race gold

Montmartre is by quite some distance my favourite bit of Paris. And yes, I am aware this is the most hackneyed, basic-bitch, Emily-in-Paris Paris opinion in existence. Yes, the hills are murder in the heat and the tourist traps stud the route like landmines and you are never more than 10 seconds from hearing an American accent, the ultimate Parisian buzzkill. But I just love the history and the beauty and the topography of the place, the ornate hum of chaos, the way the wind breathes over the cobbles, the way a world-famous heritage site is still recognisably a place where people live and convene and buy bread rolls.

Most of all, I love the randomness. No two visits to Montmartre are the same. Over there, a guy carrying a mattress down the street. Over there, a guy on a motorbike having an animated argument with a brasserie owner. Over there, a couple bickering over whether to order nachos or a burger and whether that tote bag really needs to be on the table. Over there, a 50-metre queue for brunch.

In this context, in an Olympics whose defining motif is the idea that sport and culture are two sides of the same coin, there is something strangely fitting about having the road race wind through the district of Van Gogh and Picasso. Perhaps you can even glimpse them there, on a distant balcony, watching the bikes whiz by: Toulouse-Lautrec dabbing at his easel, Matisse marvelling at the modern cyclist and his economy of form, Modigliani swearing in between slugs of absinthe that he can see three Remco Evenepoels and one of them appears to be sitting on an elephant.

There is an irony here: that for the centrepiece of the world’s most prestigious bike race, Paris so rarely gets to see cyclists racing each other. If you were an American private equity investor charged with reinventing the Tour de France from scratch, this is surely the part you would fix first. This is the climax of your big show, your World Cup final, your money shot. You’re handing glasses of champagne out of the team car? And everyone already knows who’s won? Dana, get me my executive paddle.

But anyone who know bike know that this is a sport steeped not just in tradition but in reverence. The final stage of the Tour is a celebration of the trials that have gone before it, of every rider who laboured under a time check chugging up some minor Alp, dreaming of making it to Paris. It teaches us that the glory is not in the ending, but the journey. No, keep the champagne. Keep the Champs-Élysées. Just not today.

I am a very serious sports journalist, so allow me to reassure you that I did not misread the start time on the Olympics website, did not stumble across the early stages of the race in Ivry while emerging from a morning gym session, did not leave my headphones on the metro, did not drink three pints at Corcoran’s Irish pub while waiting for the race to arrive, am not writing these words at a delightful pavement cafe while stabbing at a galette au chèvre. I am all hustle. Hang on, that sounds like the race approaching. Let me go and check who’s winning.

OK, so they all went by in a bit of a blur and the crowd was about 25 deep and it was hard to tell. The signal is a bit patchy so I couldn’t check on my phone. But as anyone who know bike will tell you, the glory is not in the journey, as some will misguidedly insist, but in the bits around the journey. The optimism and the anticipation. More Belgians than live in Belgium. The sheer countercultural pleasure of glimpsing one of the world’s greatest bike races for free. The way the creaminess of the goat’s cheese contrasts with the savoury buckwheat and the acidity of the balsamic glaze, melting and melding in your mouth in a way that evokes ... oh, busted.

We’re all friends here, so let me try to explain this to you while I sink a fourth pint and try to block out the little voice in my head reminding me that I still have to go and cover the evening swimming later. A recent study found that for the first time in decades, bike journeys in Paris outnumbered car journeys. The mayor, Anne Hidalgo, has set in motion an audacious target of making 100% of Paris cycleable by 2026. To this end, she has sanctioned more than 1,000km of new cycle lanes, 120,000 new bicycle parking spots, 1,500 new Vélib public bike terminals.

These are choices and they are not easy ones. By way of example, Marine Le Pen’s 2022 election manifesto did not mention cycling at all, although it did pledge to renationalise the motorways.

In an overheating world, in increasingly atomised societies, bicycles can be the magic bullet. Cycling is going to help us combat the climate crisis. Cycling is going to help us live healthier lives. Cycling is going to give the poorest in society an affordable means of transport and open up cities in a way we can’t yet imagine. And if we’re going to create a city of bikes, then having a bike race whizzing up the streets of Montmartre seems like a pretty good way of selling it.

Sport is not product. Sport is not asset. Sport is not a walled garden guarded by G4S. Sport is the greatest social glue humanity has invented, a celebration of the body and a celebration of those who celebrate it.

I met a couple from Valladolid who edged away when I tried to talk about John Toshack. I met a guy from Germany who had sunk a bottle of red wine. I met a family from LA, which, to be honest, killed my buzz a bit, but we swapped numbers and will try to go for dinner in 2028. All here to share in something new and stirring and visceral.

Sport is for everyone. I believed this after no pints and I believe this after four. The fragments of the peloton are grimacing their way up the hill for the third time and everyone here has been waiting for hours and to be honest I’m not sure how many Remco Evenepoels I saw. But here on the slopes of Sacré Coeur, we are all the same and we are all happy. Joined in our shared mission and our strained necks and the pure, licentious freedom of the bike.