You are correct that a probability distribution of future prices can be stated. In fact, the market prices of options on a given stock imply such a distribution -- including the tails (far-from-the-money options).
The statement that it's "impossible...to predict future stock price" is shorthand for it being impossible to do so in a way that beats the market on average. The main parameters governing the market-consensus distribution are the current stock price, which determines the mean of the distribution, and the current expected or "implied" volatility, which determines the standard deviation of the distribution. And yes, volatility is empirically "persistent", so the past "realized" volatility in a stock plays a big role in how market participants arrive at the implied volatility number.
The key point is that if the market (including the options market) is "right" about the probability distribution (i.e., if future stock prices behave as if they're drawn randomly from it, and have no further predictability), there is no way to beat the market. An investment strategy that you believe will beat the market corresponds to a belief that the market is wrong in some way about this future price distribution. It's like how betting on a coin flip at even odds can make money, on average, only if the coin is biased (the odds are wrong).
For example, if you think the mean of the future price is significantly higher than the current price (or in plain language, the stock is likely to go up), then you would overweight a long position in that stock. If instead you think the standard deviation of the future price is significantly higher than the implied volatility in the options market, you would use an options strategy like a straddle that bets on a large move but is neutral on direction.
The fact that a given stock is likely to have high, or low, volatility is only profitable information if it is not already the consensus reflected in (options) market prices.