I did my PhD studying small molecule amorphous solids.
The thing with the church windows is as you suspect. It was really difficult to make windows of uniform thickness before the float glass process. What they would do is take a giant glob of hot glass and spin it around to make a big plate which they would cut out into the desired shape. This would result in a plate that was thicker in the middle than on the outsides. In order to keep the plate from falling over, it would be installed with the heavy side down (generally). From time to time, you'll find a window that's been installed upside down and will be thick side up.
Glass is neither a solid or a liquid. The glass transition is not a phase transition (first or second order). Glasses are kinetically trapped states, out of equilibrium, and thus do not qualify for the definition of either solid or liquid as those states are only well defined for equilibrium. On short time scales, glass will behave like a solid (it will shatter if you hit it with a hammer). On long time scales, it will in fact flow like a liquid. For conventional window glass, the timescale for flow is billions of years. Pitch is a glassy material, and in that case the timescale for flow is comparatively short at just a few years. The reason pitch behaves differently than window glass is because at room temperature, the molecules of pitch are able to move much more quickly than the atoms in window glass. This is similar, but not equivalent, to saying that pitch has a lower T_g than window glass.
The flow of glassy materials is not a melting process. Pitch that has flowed out of the funnel and made a drop will still shatter if you hit it with a hammer. It is not fully accurate, but reasonably accurate, to think of glass as an extremely slowly moving liquid. If you hit glass with a hammer, the molecules don't have time to get out of the way and the piece will shatter. If you give glass a couple of billion years in a funnel, it will have flowed like a liquid.