I haven't dug into the engineering behind this yet, but my main concern with any custom Linux distribution is it often ends up as a waste of engineering.
It's pretty easy to write your own Distro and pair it down to the essentials, and pairing it down that way allows you to strip out complexity, making it easier and more reliable to patch it. But that then means you are now maintaining this custom thing forever. If you're Amazon, that might be fine, but I suspect that this will be dropped when it is no longer profitable or a competing project supplants it - meaning in 5 years this thing might be gone. (A common theme of custom Linux distributions)
And then there's troubleshooting. With a stripped-down distro, you will eventually need more tools to debug the thing, meaning you have to build and maintain packages to do that. Bottlerocket's answer to this is "run them in containers and use our API!", but I'm not sold on this. Have you ever tried to do debugging between host and container, or container to container? There's a lot of b.s. you have to hop through, and most Unix tools were not written with it in mind. I highly doubt that it will magically work for everything. If that's the case, then this "don't worry, because magic" idea is not really saving you work over maintaining a traditional OS.
Moreover, you don't need a custom distro to do live patching. There are simple tricks you can use to juggle files and processes during a live patch, to say nothing of "checkpoint process && 'mv old_file new_file' && thaw process", etc. Kernels support live patching too. So if the argument is "well it's easier to patch", i'm not sure you're not trading away "easy" in one place for "pain in the ass" in another (see above). All of this also argues that it's just as effective to treat live-patched systems as you treat immutable infrastructure, and I'm not convinced of that argument either. The former is just more complex, and complexity attracts failures.
Ultimately I think what you'll find is Bottlerocket will get a niche following, but also some people will get annoyed by it and go back to regular distros which already have well defined methods.