Why was the Davson-Danielli model rejected?
The original figure that Danielli and Davson proposed looks like this (from the original publication):



It shows the phospholipid bilayer of the membrane (which is correct) embedded between two layers of globular proteins. The hydrophobic tails of the lipids are orientated towards each other, while the hydrophilic heads are oriented to the outside.

Although the membrane composition is correct (this was already published 1925 by Gorter and Grendel), there are some problems with the proposed model:


Membranes are not identical. The differ in thickness and the ratio of
proteins:lipids.
Membranes have distinct inside and outside layers (defined by the
membrane proteins which are present on the surface of the membrane)
Other than predicted by the model, the membrane proteins do not have
a very good solubility in water - in fact they are amphiphatic,
meaning they have hydrophilic and hydrophobic regions. The
hydrophobic side is anchored inside the membrane.
When the membrane proteins would cover the lipid bilayer, their
hydrophobic regions would be in contact with water, which
destabilizes this construct. Even if they would be oriented towards
the membrane, they would face towards the hydrophilic heads of the
phospholipids causing the same effect. Additionally the proteins
would also seperate the hydrophilic phospholipid heads from the
water. So there is no real stable solution in embedding the membrane
with proteins.