It is usually stated that the implied volatility is statistically generally --- not always --- greater than the realized volatility. It seems this statement is made with regard to the implied volatility at the money or when the strike is equal to the future of the underlying. Is this statement statistically generally true for all strikes?
Most of the time, but not always. When a trader underwrites an option (selling a call or put), they do not get the choice to exercise - the buyer has the choice. So the buyer pays a "premium" for that choice, similar to the idea of insurance. If you later sold the option and it turns out that the realised vol < implied vol of the contract you bought, then you would lose money. Thus, in the long-run, you lose money when realised < implied vol.
It then follows, if one assumes that realised vol < implied vol is always true, like your question, why ever be long options if you always make money being short? Because it's not always true. During black-swan events, realised vol > implied vol. If you look at the VIX, during times when vol is > 40%, traders who are short options are generally selling their options priced at implied vol << realised vol and more often than not, would be losing money.
I go back to the insurance analogue. The insurance buyer (long) "wins" during low probability events, but is losing on a day-to-day basis. The insurance seller (shorter) "wins" small amounts on a day-to-day basis
Standard realized volatility calculation only corresponds theoretically to ATM strikes not OTMs. Unless you have a different definition of realized volatility you cannot compare it to OTM implied.
However, on the whole vol curve level I believe the statement to be true. For a pure ROC analysis you'll generally make more money shorting OTMs than ATMs.
The difference starts to occur when you look at the 3rd order and 4th order moments instead of just variance (i.e. skewness and kurtosis). That will be much higher for a short OTM position than a short ATM one.