Shorting does add to the number of shares on issue.
Here's a simple test: An investor, A, buys 100,000 shares at $1 for $100,000, and lends them to shorter B, who sells them to C at $1 for $100,000. Unfortunately for B, the share price doubles while they are shorted, and B has to pay $2 each to cover. As soon as the shares are returned to A, A sells them at $2 for $200,000. Coincidentally, C sells their shares at the same time, also for $2. A is $100,000 up, because A bought the shares for $100,000 and sold them for $200,000. C is also $100,000 ahead, because they too bought the shares for $100,000 and sold for $200,000. So while technically there was only one set of 100,000 shares, effectively there were two, and two separate investors both reaped the full benefit of a share price increase.
And if that still doesn't convince you, compare the above example with this one:
Suppose that instead of being a shorter, B was a very priveleged outfit that could just create and sell shares. An investor, A, buys 100,000 shares at $1 for $100,000. B creates 100,000 shares and sells them to C at $1 for $100,000. Some time later, B buys 100,000 shares at $2 and cancels them, having made a loss of $100,000. A sells their shares at $2 for $200,000, and so does C. For a period, there were 100,000 more shares in the market than the company itself had issued.
You will note that there is no material difference between the two examples, with respect to what is happening on the main market. Exactly the same number of shares are bought and sold by exactly the same players at the same prices and times. Shorting does indeed effectively issue new shares.
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