Why not formulate QM in terms of |ψ| squared?

[A. Neumaier]

You are obviously right, here. Nevertheless my question remains: can we quantize charge density, viewed as a field, directly?

Together with charge density, you need st least the associated charge current operaots, which complement the density to a conservation law. Quantizing this is called current algebra. There have been varous attempts and there is significant literature, but apart from the 1+1D case (where current algebra essentially turns into Kac-Moody algebra) no real breakthrough.