I don’t think large companies hire full stack developers. “Full stack developer” is just a term. A programmer who can design a website front-end, and also works on the back-end. But in reality, web development is a vast subject. The domain of web development is witnessing a growth like never before. Natureally, nobody can fully master every single language that falls under the umbrella of “web development”.
This is possibly because the scope of requirements for corporate websites has gone up so much that no one professional can manage the job singlehandedly. For example, behind a large eCommerce website, there are hundreds of developers working. Some are working on the front-end design, while others are handling the back-end and database. Specialized skills are required to succeed as a programmer. Knowing a bit of everything isn’t just sufficient.
Going back to your question, is “full stack” becoming obsolete? Not really, but it is not quite suitable for large corporations. For small companies however, full stack development is more cost effective. A full-stack developer has a better chance of winning freelancer projects. Small companies love to hire professionals who can manage everything. This is the most cost effective way of getting things done.