> People don't want stable jobs anymore.
Sure they do.
> That's why the industries where workers have a modicum of power (law, medicine, finance, tech) are dominated by "flexible" work schedules.
Flexible work schedules and stable jobs are orthogonal concerns. Flexible work schedule is “employee has some freedom to set working schedule without adverse consequence”, stable job is “employee has relative security against loss of income, benefits, and employment”.
People absolutely still want stable jobs (though they are much harder to find), whether or not they also want flexible work schedules.
What has changed is that traditional employment in the private sector isn't nearly as secure as it used to be, which makes the gig-work vs. traditional employment calculus favor the former more than it would have in the past, since the choice is between insecure work without flexibility and slightly more insecure work with flexibility, rather than secure work without flexibility and unsecure work with flexibility.
Also, for many, it's the choice between no work and insecure work, and whether or not people want stable work, they definitely are compelled by our economic system to generally prefer some work over no work.