Unfortunately Blender Render has been completely removed in Blender 2.8, so this functionality is no longer available.
Could't agree more about having to use materials slowing down my workflow when I'm going to be exporting to something that completely ignores the materials anyway.
However there are at least ways of speeding the process up, and you can still get the same visual feel as Textured Solid.
The button to the right of the colour option lets you quickly pick a new input node type while avoiding the node editor. So you can quickly switch to an image like this:
Then just select the image with the dropdown that appears.
You can get a similar visual feel as "Textured Solid" in the viewport by setting your shading up like this:
(I have both of these set up this way in my startup file. Then when I need to add a new texture, I just duplicate the material and change the texture.)