Sure, it's grammatically correct, but as said, is well out of date, and would probably raise eyebrows nowadays (in the same way that saying any antiquated phrase raises eyebrows).
Also, looking at the contest example, Person A (the referee) says 'he'. Person B (a player in the game who knows who else is competing) will be able to infer that this is meant to be gender neutral. But Person C (a bystander who does not know the gender of the players) will not infer that, especially if Person C is also unaware as to whether or not Person A is aware of the gender of the players. More so, whilst Person B would understand what Person A is saying, they'd think it a bit odd and the thought that Person A was being presumptuous as to who the winner will be would cross their mind.
In the above paragraph, I'd only write, 'whilst Person B would understand what Person A is saying, he'd think it a bit odd', if I was assuming that Person B is a male, and from this sentence, the conclusion that Person B is a male would be rightly drawn.