>are chess players required to write down their moves?

Not at fast time controls, or with little time left.

"If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then for the remainder of the period he is not obliged to [record moves]." http://www.fide.com/FIDE/handbook/LawsOfChess.pdf

And smart boards are expensive.