ugh. Ok, yes, I did it 3 times messing up the indices in 3 different ways, but I'm finally there. So this seems consistent, fine, we get what we should. What I'm still not clear about is: is this still a connection in the usual formal sense?
As far as I understand, a Levi-Civita connection is *defined* by its action on coordinate basis vector fields (https://en.wikipedia.org/wiki/Levi-Civita_connection#Christoffel_symbols), So if we now say $\nabla_i\mathbf e_j=0$, we would be saying $\nabla_i=0$. So would this be a non Levi-Civita connection or something?