Awesome! I figured it out!
Here's the Sega boot screen I was hoping to see...
I'm kind of embaressed to say what the problem was.
I didn't get an AC adapter with my Master System. The first one I grabbed that would fit was an NES AC adapter. Well, it didn't work, and I'm damn lucky I didn't ruin my Master System. (It seems the NES AC adapter outputs AC and the SMS needs DC)
I tried my Sega CD one and it works flawlessly. I'm going to rummage through the local second hand store and find a replacement Genesis model 1 AC Adapter to use permanently with my Master System.
So, anyone else that's had this problem - try a different AC adapter.