I think one of the reasons that the struggle is fascinating is because Claudius and Hamlet are both so intellectually sharp. Both are constantly mentally at war with one another, always trying to keep one step ahead...Claudius to keep Hamlet from discovery, Hamlet trying to achieve discovery.
There is no one else in the court who can match Hamlet's intellect. Polonious believes himself to be sharp but Hamlet runs circles around him. Consider the "fishmonger" scene. Hamlet dazzles Polonious with his words, toying with the man, calling him, essentially a "fool" but Polonious doesn't get it. Instead, he plays right into Hamlet's plan, that is, to convince the old man that he is indeed insane.
Rosencrantz and Guildenstern, Hamlet's childhood friends, certainly have not the wit nor the brains to pull on over on the prince. Once Hamlet gets them alone, it takes just minutes for him to get them to confess their collusion with Claudius.
Horatio is bright but never displays the kind of thinking that Hamlet is capable of.
No, it is Claudius alone who can almost play at Hamlet's level. It is he who first devises the plan to keep him close; then it is he who decides he should be put to death. It is Claudius who devises the evil scheme to have Laertes assure his step-son's death.
The play is a continual back-and-forth between Claudius and Hamlet's mind (and action) games. It is one of the many reasons Hamlet is so endlessly fascinating.