Very interesting question, Keith! My initial reaction is to wonder why you don’t just plug the consoles into your TV like you did back in the day. That’s certainly the most straightforward solution.
In any case, let’s address your specific question. Although it’s possible to connect a game console to a PC, you wouldn’t actually be playing the games “on your computer.” Rather, you’d simply be routing the console’s video output to your monitor, effectively using the latter in place of a TV.
There are a few potential obstacles here. The good news is that your particular TV tuner can accommodate your consoles’ composite audio/video connections, which is half the battle.
However, those consoles were designed to pair with CRT televisions, which offered a maximum output resolution of 640 by 480. Your monitor undoubtedly runs at a much higher resolution. It’s hard to know how your games will end up looking, but I’ll bet my lunch they won’t run at a satisfactory size or aspect ratio.
There might also be some lag between controller actions and what you see on the screen. And let’s not forget the audio, which poses its own set of potential problems.
Bottom line: computers–and especially computer monitors–aren’t a good match for older game consoles. Newer consoles with HDMI outputs pair pretty well with newer monitors (those that have HDMI inputs and built-in speakers), but even then you’re bypassing the PC entirely.