CRTs are totally cool tech, I miss working on them, it was always a rush to hear the crackle of HV coming up after a successful repair.
The 3 terminal part on the neck board is the driver transistor, it's what is translating the video signal to a level required to drive the cathode.
Check out the manual for the Electrohome G05 monitor, this is a vector monitor so the deflection circuits are completely different but the video amp or Z amplifier in vector terms is clearly described in what I consider to be a very well written manual.
http://www.arcade-museum.com/manuals-monitors/Atari%20Monitor%20TM-146%20Issue%201%20Electrohome%20G05-801%20Monochrome%20XY.pdfThe neck board should pull straight off, you may have to cut a blob of silicone or peel it but be careful because there's a fragile glass pip in the center of the pins and if you crack that it's game over. This monitor has a shield over the back of the neck board which makes probing difficult but sometimes it's not too hard to remove. Looking at the end of the CRT and starting with the first pin after the gap the numbers go around clockwise, this is the most common pinout for those small neck tubes.
1. G1
2. K
3. Heater 1
4. Heater 2
5. G1 (internally connected to pin 1)
6. G2
7. Focus
Pin 2 should have the video signal on it, don't worry about pins 3-4 because if the heater wasn't working you wouldn't have anything on the screen. Pin 7 is not important yet either. Pin 1 and 5 may be connected to ground, it is on many monitors. Pin 6 should have a DC voltage that varies with the control usually marked Screen or Sub-Brite, look around for that control and verify that it works. So here's the thing, a CRT has limited dynamic range, so if you crank up the brightness too far the gun will never go into cutoff and the whole tube will light up. This can be due to a circuit fault, or the screen control can be turned up too high. Since this control is not customer accessible it typically has a very wide range to account for variation between CRTs and if you turn it up too far you will end up with a solid glowing raster. Depending on where the diode that was shorted is it may be part of the circuit that produces the voltage that powers the video amp.
So at this point I'd say scope pin 2 on the CRT and verify that you have a video signal, let us know what range it is relative to ground, you should be able to ground your scope to that metal shield or to the ground strap over the bell of the CRT but do check just to make sure that it's tied to the earth ground first. Next check the voltage on pin 6, find the Screen or Sub-brite control and verify that the voltage on pin 6 varies as you adjust that, let us know what range it covers.