People who specialize will be better at their specialization, but you will be good enough for majority of apps.
At the end of the day, some people like Javascrip and CSS. And "full stack" only means that they will work with those, SQL, and some backend language (that may be Javascript too). It doesn't imply anything on the workload, all it does imply is that the place is hiring a generalist instead of specialists.
The problem is pretty simple. Let's say you are cutting-edge in all the technologies that make you very adaptable at time T. You get hired. You decide a stack A/B/C/... for your next project. You get stuck in that project for ... let's say 3 years. (a reasonable time for a project to reach production and have a few real-life iterations). If you are not an assh... who leaves after 2 years (leaving the production/maintenance to others, i.e never having lived the "hell" of maintaining things for real), then you are 3 years-behind in all the technologies that are not in the A/B/C/... stack.
Now, at time T+3, you are ... the obsolete man [for TTZ fans only :)]
This isn't helped by the current trend of trying to hire mostly fullstack developers, also known as hiring one person to do the job of three people. It does almost nothing but incentivize packing resumes to hopefully make the cut and burn people out from switching modes.