Constructing a Walsh diagram of AH2
I'd say the diagram from Atkins "Inorganic Chemistry" is in error.

I went to what I consider the "definitive source" for such Walsh diagrams: Orbital Interactions in Chemistry by Albright, Burdett and Whangbo.

Here's their diagram for $\ce{H2S}$ (the linear on the left, the bent on the right):



They don't readily indicate the method they used for the calculation, but on this energy scale the difference between the $1\sigma_g$ and $1a_1$ is minimal. They show that the $1a_1$ is lower in energy since the interaction with the central p orbital will be stabilizing.

As to the $1b_2$ rising above the $2a_1$ in the bent structure -- the diagram from Albright, Burdett, and Whangbo agree that there's no "crossing."