Are Adjustable Stems With Reasonable Merit?
Whether they work: Short answer yes, they do not have major widespread problems. Most problems that do occur are related to undertightening the bolts on them to start with, or adjusting them without following the right sequence (on the ones with the out-of-view bolt underneath the extension) or without proper torque or lubrication. Most of them are designed in a way where they're fairly overbuilt and will get some wiggle long before there's any danger of failure. There are some sketchy ones, but that shouldn't reflect badly on all of them.
They do add a maintenance point and some weight. Depending on a cyclist's priorities that could be a non-issue or a dealbreaker.
Adjustable stems can be a way for multiple riders to share usage of one bike and have the fit accommodate each better. The one-rider-per-bike paradigm has always been more prevalent among cycling cultures that tend towards leisure, and even there we are now in an era where some families are purchasing large, expensive cargo e-bikes for utilitarian purposes such as transporting children, groceries and home goods, commuting, etc. There is often financial and logistical/spatial incentive to buy one that can be shared among family members. Adjustable stems are not a perfect means in all cases to facilitate this by themselves (this could be its own question), since the way most of them work is that you bring the bars way in at the same time you bring them up, but they are part of the toolbox to be sure. In particular, they have limitations when one rider is taller than the other but both want intermediately extended riding positions, but where they're almost the ideal solution is when the riders are fairly close in stature but one prefers or needs a much more upright position.
As for the problem-that-doesn't-exist question more broadly: they can be oversold by manufacturers and dealers as able to solve more fit needs and issues than they're really capable of, but they can also be of use to someone that either needs a very upright position and can't or won't get it any other way, or to someone that may need to start more upright and acclimate downwards a bit and doesn't want to buy new stems along the way. People's physical needs for bike fit do change sometimes; retiree comfort bikes, for example, are a pretty appropriate place for OEM adjustable stems, in my experience.