Replacing keypad with LCD question

I want to replace an old 15 button keypad that uses 4 by 4 wire matrix (16th button isn't used) but those are by physically connecting 2 wires to make a key contact.

I want to replace that with LCD that can display keypad and use the touch screen function to pass the key press to an old system but I can't find info on how to simulate key presses using touch screen LCD. I tried Google and only found Youtube tutorial but I really hate using YT because they are not friendly with deaf people and the speech to text translation rarely works correctly mostly when odd or unusual word are used or when the speaker has accent that fouls up translator.

Is there a guide I missed somewhere or can one of you explain how to use Arduino to simulate key presses by shorting 2 wires together when someone presses on the LCD?

I want to use LCD so I can change graphic to reflect what is currently running on my Atari 5200. The controller are prone to failure (plus non-centering analog stick sucks for games like Pac Man) and my idea is to build a nice arcade style controller with 2 arcade fire buttons, a decent analog control stick (maybe a nice PC controller?) and LCD display for keypad. SD card will be used to store the images that I can choose based on which game I am playing.

There's probably not any tutorial that directly addresses using an LCD and making arduino simulate a button press on two contacts. But there are certainly many tutorials on how to simulate a button press by shorting two contacts. And there are certainly many tutorials on how to do something in response to a press on an LCD screen. If you can learn how to do those two things then putting it all together would be trivial.