MS view Achievements as valuable and protect the integrity of their “worth” very closely. i.e. they have a certain difficulty or time input to get.
In other words, developers can’t just add Achievements to any game because it wouldn’t be long before someone made a game that handed out 250 points for pressing the start button, which would then devalue the whole Achievement “economy”.
Obviously there are some Achievements that are easier to get than others, but in general some effort is required. There was a time when free games couldn’t have them but that was relaxed with the advent of Free to Play popularity.
So as well as the previous answer that Xbox games have to be certified, and that certification process is lengthy and detailed, there are also specific requirements about Achievements as well which MS check as part of the certification process.
It’s the same reason the makes of Indie Games on the 360 couldn’t add them.