Which sentence demonstrates correct pronoun-antecedent agreement?

A unique feature of man is their thumbs, which are unlike those of other creatures. Everyone should bring their own money.

Each student wrote his name on his paper.

I believe that the 2nd sentence would demonstrate an appropriate agreement firstly, because of the modal 'should' which means that not all people bring their own money, and secondly, because if not all people bring their own money, this statement implies 'plurality', and not an overall majority that could have attracted an agreement (in singular) to the subject 'everyone'.