People I know who don't, or won't, play older games tend to have three main reasons for why.
1. Graphics
The graphics are bad and they don't like it. Simple as that. I have a friend whose brother would ****ing love the original Deus Ex. I cannot stress how much the game seems almost tailor made for him. But because the graphics are so old, with blocky architecture, stiff animations, and low quality textures, he'll never touch it. Another friend of mine will never play the original Thief games or System Shock 2, even though I know he'd absolutely adore them based on his love for Dishonored, Prey, and other games of that type. He's even acknowledged that himself, but says the graphics alone are why he will never touch them. It's a damn shame.
2. Outdated Mechanics
The other main reason I've heard is that the gameplay is outdated. Things like tank controls in older Resident Evil games, or lack of dual analogue control in shooters like Goldeneye, or poor UI design in some strategy games are all things I've heard used as reasons to not play classic older games. The Secret of Monkey Island is a great game but a friend once said he'd never play it because he didn't want to have to click on a verb and then click on an object. I've had people tell me they will never play Morrowind because it doesn't have a full voice-over and so they'd have to read all the dialogue. Modern games have a lot of standardised systems, for better or worse, and there are many quality of life touches that most older games lack. Some people can't do without them now, or think the experience is lesser without them.
3. Availability
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@Shroom and
@Dizzi said, actually being able to play older games can be a hassle. Digging old consoles out of storage and setting them up is getting harder as they physically age and TVs slowly phase out the inputs they require. I remember the last time I played my SNES I had to use my Gamecube's video cable because my TV didn't have the input the SNES needed. I was impressed the Gamecube cable fit the SNES, to be honest (it also fits an N64 I believe), but it highlights one issue with older consoles that will become more of a problem as time goes by. A lot of people simply don't have their own consoles anymore, and buying a new one online to indulge in a bit of nostalgia, or to play one game can be a bit much for a lot of people. Emulation is an option, but, depending on what you're emulating, that requires a computer with enough power to do it. NES and SNES is easy. Gamecube and WiiU is harder. Emulation also puts some people off because of the legal minefield it can become and some people not trusting ROM sites. Emulators also aren't always perfect, either, and unless you have a controller, using a keyboard to play old Nintendo or Playstation games can make people not want to play, too. As for PC games, a lot of them don't run on modern hardware without patches and mods, which often have to be made by fans themselves. Arcanum: Of Steamworks and Magick Obscura is a fantastic RPG that is utterly unplayable without fan mods to make it run on modern systems in any kind of acceptable manner. Vampire: The Masquerade - Bloodlines, also one of the best RPGs of all time, is famously a horrendous mess of bugs, glitches, and unfinished content that all but requires fan patches and bug fixes. This is too much for some people, who just want to play a game and not do a software engineering project.
Those are the main reasons I've personally heard to explain why some people are against playing older games. It's a shame, really, because there are people I know who are missing out on games they would love because of the graphics or because they'd have to click three extra things to get it running.