All of that notwithstanding, it is very possible and in fact extremely common for alpha traits to be displayed in chess, both the Machiavellian style and the more “woke” for lack of a better term. In a one on one game, this is communicated in the non-game communication ie body language. In a club or team setting all of the usual social dominance tactics apply.
Honestly I dislike using the term alpha, because it's so inflammatory. Its hard to talk about what alpha means without bringing up beta, and the moment you use those two words then it sounds like redpill-speak. However, there are certain kinds of people who are confrontational and very much extroverted, the kinds who don't back away when challenged. I see alpha as a way to describe these traits as they are, with no moral conclusions attached.
For sure Chess players can display alpha traits, but it's hard to see how Chess players are MORE alpha than Diplomacy players. An entire game of Chess can finish without a single word exchanged, and it's rare and unprofessional for violence to show up in Chess, though it does show up time to time. In diplomacy, violence, albeit only verbal, is the norm. A nonconfrontational person will never survive in diplomacy, but they can do well in Chess.
For that reason, I encourage you to stop conflating "alpha" with antisocial character traits. Words matter. The very idea that this is acceptable behavior for leaders, that sometimes violence is the answer, perpetuates the social norms that allow bullies to stay in power.
Also, your conception of diplomacy needs work. Again, anybody who is actually any good at diplomacy understands how it is possible to convey a firm position with minimal violence. Diplomacy is more of a dance. You don't want to be the kind of lead dancer who's violently swinging your partner around the room.
If you really believe that confrontation can't be resolved without violence, I recommend a meditation practice known as "metta" [compassion]. Reading the Art of War may also provide inspiration and guidance on this front.