It's not "grieving", per se. You press F to touch a coffin. Just like everyone else at the funeral, presumably. Remember that one of the core design choices of Advanced Warfare is that Mitchell never talks outside of pre-rendered cutscenes. So they couldn't have you walk up to Irons and Press F to pay respects, and trigger him saying something like, "Sorry for your loss." Instead, they went for a non-verbal communication of Mitchell paying his respects to a dead friend. As I said, Mitchell is the player, and the player is Mitchell.
The issue was due to two things: verbalizing an emotion in such a way that it became laughably mechanical and forcing the player to complete that mechanical interaction as part of the game's checklist progression system without any other roleplaying. Paying respects at a funeral is to offer condolences or sympathy which are emotional responses, not simply touching the coffin which is the physical action itself.
Here are some alternate ways the scene could have worked, had more impact for players who were engaging with the story, and could have prevented it from turning into a meme:
1. Remove the text entirely. Simply have the player walk up to the coffin and display the interact key (F, X, whatever) to let them know they can do something. The exact same animation plays but without the text explicitly telling you that pressing that button just made your character express emotion. Some players would see the physical action of touching the coffin as a polite formality while others would see it as touching, but not verbalizing it removes the mechanical insincerity of telling the player that their character was emotional on command.
2. Don't even make it a button prompt. Just make it so players can observe other characters walking up to the coffin first, and then allow the player to do the same. If the player stands over it for a half second, then trigger the animation automatically.
3. Make it optional in any scenario. Part of the issue was that it was required of the player before they could progress. For players who just wanted to run and gun in a game about robots and heavy weaponry, making "paying respects" be a forced mission checklist item no different from any other mission about exploding robot heads made it less unique and more laughable.
If this was a Fallout game and the "paying your respects" was one of several options provided when you were, say, talking to the dead character's mother, it could also work. In that case the player is already roleplaying and has several choice on how to handle the situation, at least one of which will presumably line up with how the player actually feels. Forcing it as the one and only emotional response in a game that never deals with that type of situation again is part of what makes it less sincere and explicitly mechanical.
4. The worst option, but at the very least change the text to describe the action itself "touch the coffin" rather than the emotion. Pairing any simple mechanical button press with a complex emotion (press F to feel regret, weep in sadness, express remorse, etc.) is never going to hit the emotional target. Maybe that's what they intended, because within that context 'touching the coffin' and 'paying respects' can both refer to the physical action alone; however, 'paying respects' also refers to expressing real emotional sorrow as well, which is why it comes off as silly.
Additional edit: Representing a physical response with a button input works fine within games. If a drill instructor in real life told you to jump, or punch, or shoot a rifle, or use a knife, or even to say you're sorry or call your mother and tell her you love her, you could do so instantly because these are all direct physical actions that can be completed regardless of your real feelings. But emotions don't work that way. If that same drill instructor told you to feel happy, or remorseful, or melancholy, or loved, or surprised, etc. you couldn't necessarily experience those emotions instantly. Having videogame characters who can experience emotions instantaneously at the press of a button is what's so silly.
It's the same reason the Squenix
"Please be excited" line also became a meme. A person can't simply become excited if asked. People don't have direct control over what they find exciting. At best you could fake the physical appearance of being excited, but you wouldn't be able to force yourself to feel emotional excitement just because someone asked it of you.