can't get my head around is why the dividend gets treated as if it was just new money.

Because it is new money that you received because you are a stockholder. Once you have received it it can't be distinuished from the money already in your wallet.
Looking at it another way, when a dividend is paid the stock price goes down an amount equal to the dividend if there isn't any other stock price movement (which in fact seldom happens). So in your example, if you had $100 of stock which paid $10 in dividends and you were reinvesting dividends you would receive addtional shares of stock that represent the $10 so the price per share would be less but you would still have $100 worth of stock. If you were taking the dividend as cash you would end up with $90 worth of stock and $10 in your pocket for a total value to you of $100.

Bob