I think the 'weak' God is tacitly accepted by some Christians, since they don't really think that God will suddenly cure everybody in the local hospital. I suppose you get a nominal kind of belief in the all-powerful, but pragmatically, well, no.
As to the implications of this for Christian belief, I have no idea. It might become a kind of kindly ethics, with carols. It might wither away. And of course, you will get the swing to the opposite, I mean, the nutty stuff that God does cause earthquakes because of the gays, or something.
Process theology has been fairly influential, started by Alfred North Whitehead, and latterly Charles Hartshorne. God, this stuff is boring.