Yes, you're in the ballpark but you make it unnecessary complicated.
Everything can be done with one material on one plane. (2 if you need obvious visible glass)
You basically just need a material with the bitmap in the self-illumination channel and a low gloss value in the reflection tab.
For the glass plane/mesh, just a simple glass material. Some slight fingerprints in the gloss slot would be good.
If the screen needs to emit much light into the scene, create another plane and position it where the main screen plane is.
Create a light material(button) of the self-illumination channel and add it to the new plane.
With composting tags on the meshes you set the light material to be invisible to the camera, main screen material no shadows casting, exclude standard mat from light mat etc.
There are also other methods like using the rayswitch shader or having just the light material as the base layer in a layer material and mixing the glass into another layer(though the glass mustn't be transparent) using a shader. (preferably fresnel)
That should be everything if I didn't forget something.
Btw, you can try out my CRT material. There I have settings for the materials that would also fit an (O)LED Display and a separate glass material for a thin cube that can be put in front of the screen(just enable it). If you're using the standard material as the visible screen, make sure to enable the reflections channel. I think this
could be exactly what you're looking for.
https://www.corona-materials.de/en/material-library/special-effects/crt-screen/246