I believe that the red connector in this case is the 5-pin AC input connector from the transformer. You can try cleaning that connector with deoxit and see if that helps. Make sure that the pins are making proper contact inside the connector. Sometimes the 2-pin STIC wire can be a red connector too, that can come loose and need attention as well.
The junk on the capacitors is indeed glue used to hold the larger capacitors in place. If this glue has disintegrated over time, there may be some stress on the solder joints that requires reflowing.
The brown gunk on the solder side of the board is leftover flux from manufacturing. Isopropyl alcohol will get rid of it. I had one power board that had particularly bad flux residue, it's good practice to remove it.
As a matter of course for two of my main Intellivision units, I replaced the three electrolytic caps and both the 7805 and 7812 voltage regulators on the power supply board. The SHOEI branded electrolytic caps are generally very good quality but they are over 30 years old now. Replacing these parts fixed a black rolling bar picture issue on one of my units.
Your picture looks pretty good - The black shadows looks like it could be some RF interference or fine tuning. Have you tried the unit with an old CRT? Try a better shielded coaxial cable? Manually fine tuned the analogue signal a hair? A certain amount of artifacting (shadows and colour bleed to the immediate left of contrast areas) is normal.
Edited by HunterZero, Fri Jul 21, 2017 8:25 AM.