Welcome to Retro gaming hell...if you have an LCD unfortunately your woefully limited
Personally I'd look at PS2 via Component (sexy as hell) or if you can a Dreamcast via VGA (Looks absolutley lush on some games)
Both have amazing back catalogues of perfect pick up & play games & can be had for peanuts as well.
Other than that you really need to look at buying a cheapo small CRT from a charity shop for a few £'s to get the best from anything retro.
LCD isnt best I'm afraid.
Console Passion are pretty good & will mod consoles as well which is a result as you could pick up a cheap Megadrive & get it RGB & 60Hz modded which will improve picture quality slightly on the LCD & using scart should eliminate your problem with tuning it in.
Also consider a SNES & doing the same but also get it done for Multi region so you have access to US/Jpn releases.
Lastly the Saturn would be a good choice too but you need to import for this to be worth investing in.
RGB from the box, has an amazing run of games including some of the best shooters ever.
If your ever local to Basildon & fancy a taste of classic Saturn/PS2/SNES/MD/DC gaming to whet your whistle then give me a shout by all means & you'll be more than welcome
Neil