It would be a basic biologic protection system. The core would need to produce a sort of totipotent stem cells and, probably, keep the blood saturated in them. The stimulus to produce such cells would then be the depletion of such cells in the blood.
At the same time the organ has to "retire" the cells grown too old and weak, and implement some kind of regenerative check on all cell lines to prevent random mutations from devolving into cancers. Or maybe the Hayflick limit for "normal" or fully grown cells is much lower, so they don't have the time to degenerate but go into apoptosis - they live fast, die early and leave a beautiful corpse.
(So another difference of the Evas would need to be that their cells are on average much "younger" than a normal human's).
For this to work, at least two very hard bounds are required.
One: reproduction becomes incredibly complicated, unless the core allows for a "grace period" before going active (maybe just after puberty?) and pregnancy shuts down the core enough to let a foetus grow and mature to term undisturbed.
Two: the core itself cannot "self-check". Immature core cells are immature core cells, mature core cells are totipotent stem cells. In some ways, the core is a stabilized tumour, and in some other ways it behaves like a liver. It can have some limited regeneration capabilities, but the reproduction of immature core cells is a very, very touchy business - triggering it too often or too fast is likely to make the whole process go awry in some terrifying way. Most likely, the core goes fully cancerous and starts eating up the rest of the organism.
(It might be possible to have an Evangelion regenerate by pumping him full of some metabolism-slowing drug, and keeping him hypothermic and in a coma. Or transplanting a large enough compatible core chunk).