There is no good answer
Let's start with this, because it's true. There is no perfect answer. I just reread the first <PERSON> book a couple days ago, and there is no answer in the book that gives us insight into <PERSON>'s reasoning. The very best we can do is guess or make wild theories. It's important to keep context in mind, that we should be looking for reasonable answers, and basically all answers will be opinion based, or conjecture.
Out of Universe
This seems like a plot hole. One of those things that, when I read the question, I said, "Yeah... That is kinda weird. It seems like the obvious choice."
1. It may be that <PERSON> simply overlooked this option.
2. Perhaps it seemed obvious to <PERSON>. There are a lot of "rules of magic" that we never actually see in the books, like all the "complex diagrams" of how magic actually works. It may be something simple like, "Well, they obviously couldn't be the king. It's against the rules of <PERSON>'s game".
3. It makes <PERSON> even cooler. Not only does he beat <PERSON>'s chess set, but he beats her set with three pieces he cannot lose. If <PERSON> is king, it takes away from how good <PERSON> is.
4. The editors/Rowling may not have wanted to make such a cliche moment as <PERSON> taking the place of the King. Like, yeah, we get it, he's the main character. It would seem pretty asinine to make him the King piece.
In Universe
There have been a couple in-universe answers already, and I'll address them briefly.
1. They have to "Play their way across the board". This is possible, and does at least have one cited source in the material. However, it seems unlikely, as <PERSON> seems to play the whole game to win, rather than to make a path. It seems like the kind of detail <PERSON> would have added had that been the plan. 805
I think that "play our way across" was a turn of phrase, simply meaning that they have to win to go on.
2. The chessmen talk back. <PERSON> is a wizard who grew up with wizard's chess. While <PERSON>'s chessmen don't seem to talk, <PERSON> may be expecting them to give him a certain amount of pushback on any decision he makes. Therefore, he put <PERSON> and <PERSON> in important positions to help the game. This seems like it wouldn't be the case either, however, and for a variety of reasons. Why wouldn't <PERSON> make them more important pieces? The Queen, for example, if he needed troops that would obey him. If your <PERSON> doesn't listen to you, you generally can't win a game of chess. Furthermore, yes, wizard chessmen backtalk, but they don't disobey. The problem <PERSON> describes playing wizard's chess is that they try and convince him of making different moves than what he would normally. But <PERSON> wouldn't have this problem. He is a veteran player, and he wouldn't take the word of his pieces even if they did talk back.
So, what would make sense? Well, there are a few in-universe answers that might fit an 11 year old chess master.
1. <PERSON> and <PERSON> are a Rook (castle) and Bishop, on the same side of the board, and on the opposite side of <PERSON>. It's possible he planned on keeping one side of the board as protected as possible to keep <PERSON> and <PERSON> safe. Not only are they on the "safe side" in this idea, but they are two very mobile pieces. The Rook and Bishop can both go from one end of the board to the other in one move, so they can escape danger as needed.
2. The bishop and the rook were option 2. It's possible that <PERSON> wanted <PERSON> to be king, but the King is irreplaceable. The transaction where <PERSON> wanted this was cut from the book because it didn't really matter. Only the most critical SE questioners would ever really care. Logical fragments are often cut from kids books, as kids tend not to care as much.
3. <PERSON> may have cared more about his friend's safety than for beating <PERSON>. When the king is checkmated, they lose, but maybe the king gets bludgeoned to death upon losing. 194