In regards to Trust No One, it's not the first game to have codes in it....
while True: learn() has promo codes that unlock different cats, and The Caligula Effect Overdoes has world record passwords that unlock optional boss-type battles with special rewards. In both cases though, you're not meant to find the codes/passwords on your own during the course of a single playthrough. You are meant to be able to find them online with the assistance of other people.
MarkoH01: The problem is that most of the gameplay is finding out those codes and answers - so without having to search for them you basically does not have a game anymore.
And that becomes an issue. If the gameplay revolves around finding codes that also requires being able to contact an email server outside the game, I would argue that the game is DRMed in terms of it's gameplay functionality. Yes, you can brute force your way by just looking up all the codes, but you can no longer play the game as it was meant to be played when that server goes down. Nobody should trust that an email server will remain up forever.
edit - I found a walkthrough for Trust No One...
It's not even the email server thing. If you watch the full walkthrough (which is about 10 minutes), they really are in a web browser and going online at times. This included two searches on Google, including a wikipedia page, and going to the website of a fake business.
https://www.youtube.com/watch?v=q6COwMbFHFA So besides an email server not going done, you're depending on a fake business site remaining up, you're depending on Google not leading you astray, and you're depending on nobody editing that wikipedia page to update it with wrong info.