I swear, when the music started playing, I thought I was listening to Turtles in Time. Of course, it shouldn't be a surprise since Konami made both of those games. For some reason, I always confused Sunset Riders with Wild Guns.
Personally, I think using the SNES and Genesis version of TMNT is actually a better comparison. Turtles in Time is a more faithful port, while the Genesis version had less bosses and longer stages. Oh and one of the reasons they can't do 4 players is because of the limitations of the hardware. Neither the SNES nor the Genesis is powerful enough to display too many sprites on screen without slowing down, especially considering the size of the sprites on screen. You'll notice Turtles In Time never had more than 4 enemies on screen, unless they were smaller sprites like the mousers.
Well, the Genesis didn't have Turtles in Time; they had The Hyperstone Heist, which was obviously similar, but still had different levels. I guess some would say they were basically the same game, but well, not really. :/
I also don't think I ever noticed anything about no more than 4 enemies at a time. Huh. I definitely don't remember that specifically!
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Hardware limitations aside, you also have to consider that at the time, the game was probably played on smaller tv's, making a jumble of sprites on the screen... also coding multitap support would have delayed the release, and they had to consider if the time spent on a feature would be worth it for an addon not a lot of people had. They would have also had to tone down the graphics so more sprites could be on the screen.
I'm fairly certain you could count all the 4 player NES games on one hand.