People are assuming that MO and charedi are two completely separate, internally-homogenous things. But that’s not true at all. It’s more like a continuum, or many continuums on several issues. There are many people in the middle (like “frum but not yeshivish”) wouldn’t describe themselves as MO or charedi.
Even in Israel there are many moderate charedim, who work and have some secular education and interaction with the outside world. Likewise, there are many MO in the US and DL in Israel who are very frum and are influenced mainly by charedi rabbis, hashkafically and halachically.
People like to lump everyone into two categories, those with and without the black hats. But it’s not that simple.