I can understand the common explanation why you don't exercise american call option on non-dividend paying stock. You can sell the option instead of exercising because price is always greater than intrinsic value.
What I don't understand is why this logic doesn't apply to american puts on non dividend paying stocks. Why would you exercise put option instead of selling it?
It does apply to puts as well. The difference with Puts is that it it more common (though still relatively rare) for early exercise to be a better option that selling the option.
But it's still much more common for selling a put to be better than early exercise. You would only exercise a put when the "interest" earned on the option (meaning the yield you get for holding a put at less than its intrinsic value) is more than what you could earn by selling the put and investing in something else of equivalent risk. This usually only happens when the put is deep in-the-money since the time value (value over intrinsic) is less the further in-the-money you get. It also requires increasing interest rates and low volatility.
If you choose not to take the haircut, you could place a STC order for a higher price than the discounted bid. Maybe another trader or the market maker gives you a better fill but there's no incentive for anyone to give you the full intrinsic value. While waiting for a better fill, the underlying could change in price and your option could lose value.
To avoid the haircut, do the discount arbitrage yourself. In the case of a call, short the underlying first and then exercise your call to close the equity position. For a put, buy the underlying and then exercise the put.
