Ah, OK.
So we agree that “he or she” is mostly unnecessary. although, ironically, right before I read you post this morning, I made another and used “he or she”, in the following context: the addressee was clearly a woman, judged by the screen name, but in my post I was making a brioader reference, which, however, was pointed at her. Hence I couldn’t use “he” alone.
I think structuring the phrase for plural is fine when it comes naturally, but people should not strain to do so. “If they participate in the raffle they should be proud” is fine. “They who win this raffle will have a nice prize” is strained and misleading, the winner is usually one person. “The winner of this raffle will take a grass mower home with them” is definitely not fine, it violates the English grammar.
In the example you gave, I would have said "Whoever wins this raffle will have a nice prize" and in the second example, I would personally have just left the last two words off. Problem solved. However, I am beginning to hear things like that last example a great deal more, and although it is grammatically incorrect, it sounds very natural to me.