In short, games are optimized and created for the hardware that they are expected to run on and when we do not know better, we hardly notice.
The long answer is:
The "old games" were created when input and screen were more or less lag free, thus there were hardly any measures necessary to alleviate the negative effects of lag.
Nowadays games are created with lag in mind. Jumps in platformers still count for a limited time even after you are over the edge, which - named after Wiley Coyote from the cartoon Road Runner - is refered to "coyote time" if I am not mistaken. There are a lot of assists in games to smoothen out your experience, like in GTA V when you walk around a car the AI takes over your movements partially so that you do not constantly bump into the car and get stuck.
Here an article that talks about a "test" of input lag in some modern games:
https://www.resetera.com/threads/ea-mot ... etc.80420/. "Test", because he is eyeballing the frames with a counter on screen and knowing to what lenghts our corner of the gaming culture goes to test lag properly, this is a very rough lag test at best. None of these games will have only one or two frames lag when your average TV adds at least one frame and the gamepads have a few miliseconds of lag as well. Note that even a Sega Saturn that was lauded as an excellent Shmup console has 3 frames lag on a CRT according to the link showing the inherent system lag of a Nintendo Switch later in my post.
Here my assumption as to why we do not feel the lag: The brain is a huge deceiver and creates in our mind what is expected. If you stay upside down for a while your brain will flip the world around again, because this is the expected sight. Similarily when you press the button on a gamepad you expect an instant reaction, thus your brain creates this impression that it is instant within a certain limit?
You can test this by playing with lag for a while and then switching to a system with no additional lag. For example, play ACA NeoGeo Blazing Star on Nintendo Switch and then change to Blazing Star on MiSTer. At first it feels like the game reacts before you press the button and the longer you pay attention to how well it reacts on MiSTer, the better you can make out the few frames of lag on Switch.
The Nintendo Switch has many retro releases, which is great, but it has inherent system lag as you can see here:
https://electricunderground.io/shmup-in ... -database/ Playing Dragon Blaze with seven or eight frames of lag? That you notice.
This is what I mean with "when we do not know better, we hardly notice". Well emulated and near lag free retro games are so much fun, because they react so well to our inputs. It feels like you are truly in control and it is your skill that counts, but to know the difference, you need to experience it.
This is why the preservation of the original experience (which includes the accurate, "poor" visual fidelity) and in general the recreation on available hardware is so important. If we can show the next generation of players how much fun near lag free gaming is (where the games demand it), then they will know and notice when lag increases in games, like with streaming services (e.g. Google Stadia).