"Off Road" has many connotations my friend. The answer you seek is murky. I could safely say that 95% of people going on a dirt road wouldn't have a problem with P tires. Conversely, I could say that 95% of the people driving through the rough deserts of Nevada would have a problem with P tires. You need E out there.
You just have to choose what off road you really mean. I run P tires but am rarely in a place where E would be better. We have dirt and little dull rocks here. Out west they have sharp rocks and thorns.
Also airing down is another consideration and I'm not talking about beach sand.
And the biggest consideration is expense. Initially you will pay 30-50% more for E versus P. In addition you will lose between 5% up to 10% efficiency, especially if you drive around town a lot.
Example: You drive 15K/yr and a lot of that is regular roads. You lose 2mpg because you with up size E.
You will spend an additional $200 a YEAR for the privilege of running E tires. Its not hard to change a flat every now and then, especially for that money. I figured I have saved around $2500 in the past 10 years running P tires. With that said, if I were off roading in certain sections in the US I would be running E tires as a rule because P just wouldn't last.