Selfhosted

If I need to do an emergency boot from a USB stick to repair something that can't boot, which it sounds like is what you're after, pretty much any Linux distro will do. I'd probably rather have a single, mainstream bootable OS than a handful.

I'd use Debian, just because that's what I use normally, so I'm most familiar with it. But it really doesn't matter all that much.

And honestly, while having an emergency bootable medium with a functioning system can simplify things, if you're familiar with the boot process, you very rarely actually need emergency boot media on a Linux system. You have a pretty flexible bootloader in grub, and the Linux kernel can run and be usable enough to fix things on a pretty broken system, if you pass something like init=/bin/sh to the kernel, maybe busybox instead for a really broken system, and can remount root read-write (mount -o rw,remount /) and know how to force syncs (echo s > /proc/sysrq-trigger) and reboots (echo b > /proc/sysrq-trigger).

I've killed ld.so and libc before and broght back systems without alternate boot media. The only time I think you'd likely really get into trouble truly requiring alternate boot media is (a) installing a new kernel that doesn't work for some reason and removing all the old, working kernels before checking to see that your new one works, or (b) killing grub. Maybe if you hork up your partition table or root filesystem enough that grub can't bring the kernel up, but in most of those cases, I'm not sure that you're likely gonna be bringing things back up with rescue tools -- you're probably gonna need to reinstall your OS anyway.

EDIT: Well, okay, if you wipe the partition table, I guess that you might be able to find the beginning of a filesystem partition based on magic strings or something and either manually reconstruct the partition table or at least extract a copy of the filesystem to somewhere else.