From what I understand, Kobayakawa was the crucial hinge of the Western Army's battle plan, expected to swing down from the right flank into the rear of Tokugawa's forces. However, other lords in the Western Army knew about his conflicted loyalties. Why did Ishida not place Kobayakawa's division where it could do no harm, and trust the right flank to a more loyal ally?
I finally found an acceptable source! Now I can post the answer! Yay!
The answer is that Kobayakawa Hideaki wasn't given the position on Mount Matsuo. He took it. According to Ishida Mitsunari's letter to Mashita Nagamori, dated 12th of the 9th month (three days before the battle), Ishida said that in the castle on Mount Matsuo should be placed the contingents from Chuugoku, ie the Mori contingents. I'm not sure which one Ishida had in mind on the 12th. Terumoto never left Osaka, though Ishida kept asking for him. Motoyasu and 15,000 men got held up besieging Ootsu castle, which wasn't supposed to happen and is probably the most likely candidate IMO. Hidemoto would arrive for the battle and be placed at Mount Nanguu.
In any case Mount Matsuo was originally garrisoned by a small daimyo called Itou Norimasa, who was probably supposed to be there until whichever Mori came and relieved him. Kobayakawa arrived and basically kicked Itou out. This was on the 14th. The 15th was the battle.
Source is the Beppu university thesis here. It examines, among other things, the nature of the position at Mount Matsuo and why Kobayakawa decided to place himself there. It also includes in the appendix a huge number of letters and correspondence relevant to the Sekigahara campaign.
After participating in taking Fushimi Castle, Kobayakawa was actually ordered to attack Tsu Castle in Ise Province. He instead marched to Oumi. The thesis argues that at that point, both Ishida and Ootani Yoshitsugu began doubting his loyalty and even talked about attacking Kobayakawa. After consulting his retainers, Kobayakawa decided to march to Mino. The thesis argues that he took position at the castle on Mount Matsuo to protect himself from the other Western forces, instead of to participate in the battle. It's possible I suppose, considering the timeline that the main forces of both armies did not begin marching towards Sekigahara until nightfall on the 14th, so the decision to fight was probably was made after Kobayakawa took his position, and maybe even influenced by it (though I don't think we're told the time on the 14th he took the position on Mount Matsuo).