First of all, the origins of this party is the "Fête de la Confederation" of 1790, commemorated for the first time in 1880 under the 3rd French republic needing a symbol to re-unite the peoples.
Being Dutch as is your punk-tattoe-pro physiotherapist (but instead merely growing some grapes, raising wines); these three words are symbols; a lot of people practice them in real life, others don't. If they were just hollow principles that guy would never have been admitted to the French healthcare system
You do have a serious amount of freedom in France, yes there are rules, laws, etc. and that might not suit you when your of the anarchist persuasion.
But I agree with your physio: my adagio when in local council is always: "Apparently all people are equal, but most of them are more equal than the others" (which doesn't give me brownie-points).
Your "independent" friends are however not showing the faults of a system; they actually need the system for the future and well-being of their kid (and the opportunity to criticize it). If not, they would have escaped to a country like Mozambique or Eritrea: everybody for themself. It's easy to criticize a system when you're a part of it and enjoying the benefits.......
Brotherhood is something you create yourself, by actively engaging yourselves in the community, with friends, politics etc. This is in no way a one-way-street. (which I suppose you already know given your academic background ;-))
I won't turn the TV on tomorrow, neither for the parade, nor for Hollande's n-th explanation "pédagogique" (does he really thinks that explaining it again and again will make the peoples accept it?) of his disastrous politics.
I will take my kids to the park on the banks of the Charente where they can play freely with all the other kids, not being bothered about where these kids came from, the color of their skin etc, and probably inviting them to our blanket-in-the-shade to share the "gouter".
Liberté, Egalité, Fraternité is neither what I think of France, nor what others might think of it. In the end it's what I try to install in our kids....