It's funny what different people find interesting in Alpine. I started using it expressly because it uses musl. Void is a distro with similar principles that offers a glibc version, so I use that when I need GNU.
Without shared linking, in your example you end up with a copy of glibc and whatever else in every process. Without some fancy page deduplication scheme (that would come with its own set of problems), the kernel cannot know that all these pages in all these processes are the same. And they might not even be, because different versions and just the process of static linking itself might make an arbitrary number of them look slightly different.
Not every program needs all of glibc, so not all of it will be paged in, but still, it looks like you will pollute your CPU caches with many copies of essentially the same thing.
I thought about whether there was some quick evidence already, but nothing comes to mind (maybe someone's less blind to it than me now).
Maybe you could take some self-building Linux distribution like Gentoo, and hack it such that it links everything statically? Compare that against an equivalent unhacked Gentoo-installation by doing a few tasks, measuring wall clock for a good first estimate?
No idea how hard that would be in practice, especially with people mentioning that glibc does not really support that. So maybe it's actually easier with one of the BSDs, e.g. OpenBSD, which is also feasible to build completely from source because it is very self-contained, depending on how that stomachs static linking for everything.
I'd also not do it in a VM. While that's not necessarily completely invalid, it does seem to skew things pretty significantly, what with at least one whole other OS running as the hypervisor and all that.