That gets complex.
It is not always a matter of inputs, sometimes resolution matters as is often the case with the NES.
And you can also do things like mod consoles output RGB.
A choice videohttps://www.youtube.com/watch?v=niKblgZupOc
and a choice linkhttp://bogost.com/games/a_television_simulator/
and while I am on a rollhttps://www.youtube.com/watch?v=Ip6WuOvK8EU
5 years ago I was getting wonderful CRTs in second hand shops, provided you had the space and the van to go grab it. Today it is harder both due to the time and due to people like me, albeit with the space, buying them up as this retro lark is back in again and not just the domain of cheap bastards like myself. You might get lucky and find that 40 inch widescreen with component inputs though (I certainly passed up a few back when) as it might be different around you.
I assume you tried to degauss your TV as well, might take some fiddling (hopefully it is just a menu option) but should be possible. Localised discolouration is usually, well when I have it then it is because I have put a magnet next to it but the classic one that most people get is the screen is in need of degaussing or it has suffered some kind of screen burn.
To answer your specific questions
The NES thing is most commonly resolution, oh and light gun does not work (when does it ever?) on LCD, that and scaling and latency but you can hopefully work around that by a choice model of screen and making sure you are in game mode (though that might in turn make for some odder upscaling).
Yes other systems will have issues, see the things I linked before for more technical discussion and yes the artists of the day knew the limitations and played to them.
The Atari line has some of the clearest examples of these techniques, see the middle link for some great examples, but whether you would quite go to unplayable in all cases I do not know.