I am most definitely a fan of continuous delivery (CD) and the idea that we have the ability to both release our software at any time, for any change, but also have the process and mindset in place to rapidly push enhancements and patches through our development process. However I'm also no naive enough to think that the processes of Agile, Lean, Scrum, Continuous Integration, or any other methodology make this happen. It requires belief and effort by your entire team, including management, to evolve your software quicker.
There is a downside. The faster, and more often, you release changes to your production system, the more likely you are to cause an issue. This isn't really any different from traditional waterfall processes, which also cause issues in production, but it does mean that the perception of issues is a bit changed. People expect higher quality when they implement a CD process, and while quality might be higher (there is debate on this), it certainly isn't perfect.
Walgreens has implemented a CD process, and they rapidly deploy changes, but they've also had issues. In fact, there's a piece that talks about an issue during a database deployment, and offers some guidance on preparation to deal with problems in your deployment process. The ideas of monitoring (which is critical), planning for recovery, coordinating development, etc are all things that we need to do in any production environment, but they are especially important in a CD process where people expect changes quickly. Whether those changes are enhancing the application, or rolling back a problematic deployment, they need to happen quickly.
Over time, which may be decades, I expect that more and more organizations and end users will expect patches and changes to their software to happen in the background, while the system is being used. This might mean we need to plan for application architectures with sections that can be upgraded separately, or maybe we will develop ways to modify executable code without ending the process. In either case, the ideas behind CD will become more and more useful, and the more you understand them, the better off you'll be.