Why won’t the Genesis turn on?

@jakesrepair the main concern I have is that you are using some sort of universal power supply. It is of absolutely vital importance that you ensure that you are using the right voltage and the right polarity. The Genesis Mod. 1 use a reversed polarity power supply. The barrel on the outside is positive and the center pin is negative. Anything else can cause a major issue.

You want to check all the diodes, bridge rectifier and replace all electrolytic capacitors. Clean your board properly and check for corrosion. Ultrasonic cleaning would be the best, but if you do not have that available, it is okay to clean with a high concentrate isopropyl alcohol (97%+) and a soft brush. Check on something like this guide Repairing iPhone Liquid Damage - iFixit Repair Guide for the cleaning. Even so it was written for an Apple iPhone, the principle is identical to your Genesis.

Here are the schematics that might be helpful getting it back to better than new condition Sega Genesis Schematic Diagrams

If you are stuck with anything or have questions regarding components etc. on the board, use this guide Adding images to an existing question - iFixit Repair Guide to add images to your question. That way we can see what you see.