Bob made a good point in finding a hole in your proof. But I'll add on some more points here.
If a stock is bankrupt and S=0, and the Put Price is K-S, both selling the Put in the market as well as exercising represent an instantaneous opportunity. You would much rather sell the Put rather than exercise, because selling takes less hassle and doesn't have to deal with the brokerage/clearing house and what not.
But you see this comes down to "real world" and "risk-neutral world" probabilities in a sense. Even though you would much rather sell because it's "easier", you have no other choice but to exercise because like Bob said who would want to buy the Option from you whilst they gain interest off the cash instead? Maybe under "real world" probabilities you could find some unsuspecting buyer, but we won't count on that.
If an American ITM Put Price is smaller than K-S, that is not probable to happen in an American Options market with early-exercise because that represents an immediate arbitrage opportunity as you already said. You would buy the Put below Parity, buy the Stock, and immediately exercise.
Playing devil's advocate, you could also buy the Put, then hold and hope that someone brings the price back in line then sell to the bid at Parity, but that wouldn't make sense when you could lock in an immediate profit by exercising it.
On the matter of the early exercise of American Options, whether a stock has a dividend or not, is generally important only when/if you have an ITM Call and the dividend value is larger than the time value. If the dividend value is larger than time value, you would exercise the Call before the Ex-Dividend date. In the case that it is not, you would sell the Call instead of exercising it.
And generally, you are always better off selling an Option rather than exercising it due to the remaining time value. As expiration approaches, time value becomes negligible and exercising an Option will be able to complete your goal with less hassle and amount of transaction as opposed to selling the Option in the market, then buying or shorting the stock.
Therefore since the time value decreases less and less as expiration approaches, you will definitely be better off selling the Option, up until the point where the time value is negligible and you would exercise it then. Otherwise you'd be leaving money on the table in both cases of owning an ITM Call or Put before expiration and not exercising.
Only exercise an Option when it is trading exactly at or slightly above Parity with little time left.