You'd have to reconfigure the kernel specifying adequate options and probably point it to the files unpacked from core.gz.
Most likely you will have to search for (kernel) documentation how to do such exactly beyond from Core, I've never seen that discussed here and there is nothing specific to Core about it, though Core may be an ideal candidate for inclusion in the kernel, as its whole base system is contained in a cpio archive.
My guess would be that a reason this is not more popular would be the lack of options to upgrade or make dynamical changes to base without recompiling the kernel.