... The love of his life just got shot. Remember that Count Saazbaum (the count he shot) had decided to take over Earth for the people of Vers, but decided to exterminate all the Terrans because they killed his fiancé during the first war. There is nothing in the act of shooting Inaho that needed to be sensible. Legally we would call it a crime in "the heat of the moment."
No, actually saving the Count made total sense. It was a survival mechanism. Not saving him would have made Slaine what essentially amounted to a Ronin; a samurai warrior with no master. The prospects at that point were not pretty, either in the short or long term. The Princess was still alive and the only person with the resources to save her life was the Count. The Count agreed because he knew that he would die, his work incomplete, if he refused. And since Slaine is by no means loved by the other counts, they would probably try to kill him. And lo and behold, one actually did try to kill him, albeit in a formal duel.
Slaine is of two minds. On one hand, he loves the Princess and would do virtually anything to make her happy. On the other hand, whether he likes it or not, he is the Count's heir and inherited not just his possessions, but also his crusade. That is, to liberate the people of Vers from their poverty by seizing for them the resources of Earth. To Slaine, the cause for which he fights and the vile nature of the Terrans outweighs the Princess' desire to see peace between Earth and Mars. It can be called a case of "Oh, she is perfect except for this one ideological flaw of hers, but that is alright because I am handling the situation."
How he pursues the war is up to him, be that aggressively or passively. Neither is really wrong. The only thing he would be in a rush to do is kill Inaho since Inaho is literally the only threat to his power. Nobody else, besides Slaine himself, has actually managed to defeat a Versian Kataphract.
As a character, he is stronger than anyone else in the anime. Period.
Moreover, he has an interesting and dynamic character. Coming up as the servant of an abusive Count Cruhteo, he showed a zealous faith to Princess Asseylum, killing a knight who betrayed her and even speaking of the Princess' will as if it were righteous; her retribution divine. That isn't loyalty that can be ignored, even by the Count, who realizes right before he dies that Slaine's loyalty to the Princess rivaled his own, if not completely outstripped it and so earned Slaine the Count's respect before the Count died.
Slaine's relationship with Count Saazbaum is even better. He hated the Count for trying to kill Princess Asseylum (twice, both indirectly and directly, chronologically), but acknowledged the need to keep the Count alive if he was to save the princess. Moreover, he acknowledged the righteousness of the Count's crusade. In the end, Slaine actually came to embody Count Saazbaum's ideals, no matter how much he hated the man.
He killed Count Saazbaum for trying to kill Princess Asseylum, even if he seemed to have forgiven the Count's first attempt on her life because of his righteous cause. He even seems to have acquired the Count's acceptance, if not approval, for the action because one has to acknowledge the parallel nature of his rage against the Count for his crime and the Count's rage against the Terrans for killing his fiance.
Slaine also took up Count Saazbaum's crusade for the poor and underprivileged of Vers. It is quite clearly pointed out in his conversation with Harklight at the end of Episode 15, the same episode which he kills Saazbaum. It is even implied that Harklight was involved in planning the Count's death. Here Slaine asks Harklight and Harklight confirms that he was born into the third class, where his poor parents worked hard to get him the opportunity to work with the Orbital Knights. Harklight admits an admiration for Slaine for rising so far in the ranks, despite his Terran birth. Slaine replies, "This new era will be for people like us. Let us get started, Harklight." Later he even gives Harklight the title of knight and a Kataphrakt of his own.
As for the Tharsis, I think you are referring to information after Episode 20, so I can't comment except to point out that Karaphracts with legs when they have hover technology doesn't make sense either, so there. In fact, making mecha doesn't really make much sense at all, since all they are at the end of the day is an elaborate firing platform. You know, like a tank, only with more guns and a lot of moving parts that are difficult to make and not worth the effort when it can fly.
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