I'm used to seeing them "starred" in an algebra 2 or "college algebra" course. Starred means the extra topics that very advanced classes could cover. My experience is even for those classes nobody bothers. I have taken a huge amount of engineering and science courses and never really felt the loss of these techniques.
The only time I solved cubics was in some of the harder AP chemistry stoichiometry problems where a cubic resulted in equilibrium problems. and then I just got x by approximation (guessing and revising guesses after feeding the number in, mechanically without a program or algorithm). And this worked fine for those chem problems (for example to get pH to single character after the decimal).
What you do see a hyooooge amount of the time is the quadratic. bsq-4ac needs to become iconic. It is in a gazzillion chem rate and equilibrium problems and it is all over physics and engineering. Either directly or in solving the characteristic equation of the 2nd order constant coefficient homo diffy Q (EE, controls/feedback, harmonic oscillator, quantum mechanics, etc.)
So I can tell you that I have "needed" the quadratic a metric gazillion amount of times. And the cubic almost never (and worked around by approximating). And really never solved a quartic.
Now I know you were asking about kids in math, but I just wanted to discuss why we don't bother with it in algebra 2 (where it sort of belongs by history and by topic). Because of course, the needs of math undergrads are irrelevent to a general algebra 2 class because these kids are a tiny fraction of kids in a high school class.
Now. The question becomes if you need to cover that stuff in other math classes. Personally I doubt it. Do you really encounter the need to analytically solve cubics or quartics in upper div math classes? No. And do you need it if you get a job that typical math grads do? (Actuary?) Probably not.
And in the tiny tiny fraction of the time that you do need it, you can just look it up in the "starred section" of your old book. Provided that you have strong manipulational muscles, you can just apply the formula for your specific problem then, without any qualms. But you really don't need to learn/remember it like you do the quadratic (not just knowing it, but using it readily).