You don't exchange any stock when you're paid dividend. If you sold a bit of your holdings each year you would eventually have no equity left.
The difference is that if the company pays dividends then the stock price will drop because it represents a claim on a smaller amount of assets. So you can achieve exactly the same effect yourself by periodically selling a small percentage of your holdings. It's like having a big piece of a small cake versus a small piece of a big cake: if you get the same amount of cake in the end then who cares how it's shaped.