I am not aware of any Linux distribution that has no package manager (except for LFS, which isn't a distribution per se). Source-based systems typically have some kind of package manager. Gentoo: portage, GoboLinux: Compile, Slackware: pkgtools + sbo, Sourcer/SourceMage: sorcery, Lunar: lin, CRUX: pkgutils, and so on.

Edit, additionally, yes. Such distributions are suitable for essentially any task for which any other distribution is suitable. The primary caution that I would offer anyone when venturing away from RHEL/Debian is how much you trust whatever organization to still be around in five years. If you're comfortable managing everything yourself, pick whatever; otherwise, it's "safest" to choose something that either contractually will still be around (RHEL), or is governed in a way that ensures some continuity (Debian).