Battle of High Hill

So I'm a decently new DM about to run the battle of high Hill for my players next week and had a couple questions about how other DMs run it.

Firstly, how did you work around the parties weapons? The book states that "Participants are encouraged to use their own armour, but are given wooden spears with padded tips to prevent them from dealing damage". So martial characters have no way to deal damage from the start of the battle until they scavenge a weapon? I was thinking it would cost an action to remove the padding, but I don't think they would just strap a pillow to the end of a spear and call it a day.

My other question was regarding the warhorses, one of my players will absolutely try to claim one of the warhorses if they survive, and was wondering about your thoughts on if this was a good idea? Is it much of a power jump? I'm inclined to let the player try and do some animal handling or something but has anyone else had a similar situation?

Thanks!