AROS can get you most of the way there, in that you can do graphics backends for it that run "rootless" on a Linux desktop, but the problem with that is still that there is no direct equivalent to the Amiga concept of screen's in X. Desktops is almost but not quite there, in that an application expects to be able to open them and place windows there. To get that to work smoothly you'd probably at a minimum need a special window manager and patching AROS to talk to one.
But the people buying this hardware are those who are not interested in / satisfied with emulation.
There are even 'AROS on Linux' distros:
http://vmwaros.blogspot.co.uk/2016/11/native-or-hosted-on-li...
TFA argues that the experience of running emulated software on this new "Amiga" is substantially better than running it via UAE on Linux/Windows/whatever, though. (For what they're charging for the new hardware, I would say it'd better be.)