HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

You still haven't answered what the difference is between recording moves in chess and here. Why is it necessary there, but in Heroes it affects the gameplay?

In chess, recording is essential because controversial situations can arise (someone knocks pieces off the board). In Heroes, players also used to manually save games (record matches) until autosave was introduced. But writing down in a notebook where a particular griffin flew means getting distracted from other aspects of the game. And if you can record moves and use them to your advantage, then that's your skill, and you're doing great.

I'm not arguing, having an initiative bar would be convenient and comfortable (like in World of Xeen with the ability to measure distances with millimeter precision), but it's no longer necessary. In the pursuit of comfort, you can reach absurdity. IMHO.