Classically ‘he’ was used to be more universal. It was always understood to include women as well. But that language is frowned upon. When I read ‘they’ I think plural people not singular like ‘he.’
When speaking about "the player," you're not just speaking about one player, but anywhere from tens to tens of millions of players. That's a "they." And as they're all acting individually and independently within tens to tens of millions of individual game sessions, it makes sense to treat that "they" singularly.
So from that perspective, it's not really a desexed reference, but a reference to a multitude of people. But even if you're talking about the person hearing a Catholic confession, and it would always be safe to refer to each instantiated member of the class as "he," "they" still makes sense. They are a bunch of priests.
I'm sure you're aware that it's usually obvious in context, like many ambiguous uses of language. For example,
> Every time the player has to restore a saved game, or pound [their] head on the desk in frustration
Changing "his" to "their" doesn't make the phrase any more difficult to read, imo. Maybe it's just something you'll pick up with practice.