Chances are the board is basically just a breakout for that controller chip. Docs are readily available for it (google the part number). You probably don't need any info from NEC to hack it.
The simplest thing to start with would be to just yank the I2C EEPROM off the board. It looks like that'll chuck the controller into a usable generic mode, removing all the NEC features entirely. It'll still have an OSD, but it may behave differently. Be warned that it may not give you the exact timing the panel wants by default, so it may not work at all.
Failing that, you can take two routes: attempt to patch the firmware NEC provided or write your own. If you can find where the relevant parts of the NEC firmware are, it should be pretty easy to just patch out that auto-OSD thing, otherwise you may end up just writing your own since what you want to do is pretty simple. The firmware is stored on the I2C EEPROM. The datasheet says that they recommend Keil's compiler, but they don't actually identify the CPU architecture used for the onboard MCU. It's probably 8051, but no guarantees. Disassemblers and assemblers for 8051 are freely and readily available. If you want a C compiler, sdcc apparently works, but I don't know of any "good" free ones (gcc doesn't support it, and I don't think LLVM does, either, so no clang).
The OSD is fully under control of that micro, from the looks of the datasheet, so you should be able to make it do anything you want. They reference a couple of custom tools (G-Wizard, OSD Workbench, and G-Probe). You may be able to get them from Genesis if you ask nicely, but you're probably on your own. Some reverse engineering is likely required.
If you can figure out the format of the OSD data, you may be able to just make the relevant graphics fully transparent or just a 1x1 pixel (which wouldn't be too obvious) or similar without patching the software at all. Of course, you'll have to find where it is in the EEPROM image, first.
FWIW, lots of modern arcade games (which are generally PC based) make no effort whatsoever to hide any of this stuff. Plenty of them will show BIOS startup text, Windows/Linux bootup, etc. when powered on, and lots of modern arcade monitors have on screen display, though the game usually doesn't change video modes on the fly, so nothing will come up during the game. You could get similar behavior here by making sure that everything you run runs in the same video mode; it's not like you're going to get native video for any classics on this thing, anyway.