In my
practical experience the set of things that can go wrong if you link apps dynamically is much larger than the problems that arise when they are statically linked.
For one, it is more complicate to keep track of which of the many shared libraries on a typical system are used by which application. It is common that the same library occurs multiple times in different versions, built by different people/organizations and residing in different directories.
Quick, without looking: which TLS library do your network exposed subsystems use, which directories are they in and where did you install them from. When you do go to look: did you find what you expected?
Have a look at all the other shared libraries on your system. Do you know which binaries use them? Do you know which versions of which libraries work with which binaries? Do you trust the information your package manager has about version requirements? Does it even have that information?
Then there's the problem of what happens when you upgrade. The servers you run might have a rigorous battery of tests. But now you are running them with libraries they were not tested against. Sure, most of the time it'll work. But you don't know that. And you have no way of knowing that without downloading, building and running the tests. Or have someone else do that.
I've been in the situation where someone inadvertently updated a library in production and everything came crashing down. Not only did it take down the site, but it took a while to figure out what happened. Both because the person who did it wasn't aware of what they'd done. And the problem didn't manifest itself in a way that made the root cause obvious.
The clearest risk with statically linked binaries is if they are not updated when there is, for instance a security problem. But in practice I find that easier to deal with since I know what I'm running, and for anything important, I'm usually aware of what version it is or when I last checked for updates/problems.