I'll vouch only for the consoles that I own(ed) and tried in an emulator. (I have never really seen the need to try the older 2600/5200/7800 games in an emulator, if anything I would rather try the XE or later series).
I found very few GG games of interest worth playing in an emulator. Largely because I feel that many of the GG games have much better counterparts in their home console or arcade versions that the effort of dealing with, what is essentially, a trimmed down version of the game not worth it. If anything, opt for the SMS or Genie versions of these games, but that brings up a different point.....
The Lynx has a few "exclusive" titles that I feel are worth playing or at least taking a look at. I wasted many hours as a kid playing games like Zarlor Mercenary, Kid Viking, Todd's Adventures in Slime World and Gauntlet: The Third Encounter (which still stands as one of my preferred iterations of the series). There are also a number of fan made games still being released, but I don't know if you can find them in ROM format. AtariAge would be the place to check for that. However, the Lynx screen is anemic at best and the games are going to look like utter ---steaming pile of meadow muffin--- on a big monitor. It does, trust me. You can make it look somewhat reasonably nice but ---steaming pile of meadow muffin--- is still ---steaming pile of meadow muffin--- no matter how much you cover it in candy. Believe me, I'm a big fan of the handheld. But if you've got the 2600 going, the Lynx can't possibly look worse than that. So it might be worth a try for you
The B&W Gameboy games seem to play alright. I think it helps that they're B&W and they seem to scale better. I haven't really tried the color games outside of the Pokemon games and I'm not a big fan of the game series so... YMMV.
That brings me to my last point... why?
I'm a firm believer in the camp that desk games, couch games, portable games and arcade games belong squarely on the target they're created for. As much as I try to engross myself in it, I find it difficult to play console games unless I'm reclining in a big comfy couch and a controller in my sweaty hands. Something similar goes for portable gaming. Arcade games need to be wrapped in a big wood (or
whatever) box surrounded by other people.
That's just me, do what you want really.