Looks at pbj's responses. The pi isn't an answer because you don't just boot it and it works and it doesn't startup or shutdown quickly enough. We've been able to get roms on the bigscreen for decades now.... the thing is getting them there effortlessly, quickly and cheaply.
Actually, you’re wrong. You can boot a Pi and have it just work, if that’s what you want. There are literally dozens of pre-built Retro Pie images all over the net. All you have to do is download one of them, install it to an SD card and you’re ready to go. That’s about as effortless, cheap, and quick as you’re going to get.
If you want to do the same thing with the NES mini, you’ll first have to work out how to increase the flash memory, and attach generic USB controllers as not all games will work well with the NES ones. So, as a minimum, you’ll probably need to hack in a USB hub, and a USB flash drive. That means getting out your soldering iron. Even if you manage to do that, you’ll still be left with a device that has only 1/4 of the RPi’s RAM, and no wired or wireless networking support. You could, I suppose, add a wireless dongle to get networking, but it’s now starting to get complicated and expensive. And once you’ve sorted out the hardware, you’ll still need to install the software.
Personally, I’d prefer to start with a device that was designed to be hacked in the first place, and which has five years of support behind it.