My favourite coats for mid-range temps give me the option. On or off, depending on weather conditions.
But the hood on my coldest-weather parka is not removeable and I would not have it any other way -- it keeps me SO warm and dry! Much warmer than a hat (though I wear a hat, too, doubling protection.) I wore that parka for long walks in frigid temps (in Ontario, not here in BC) and it was a game changer!
With short hair, a hood can insulate the neck even more than a scarf -- though again, I almost always wear a scarf, as well!
Several of my rain coats also have non-removable hoods. Again, they have proved their value. I usually wear a hat but the hood keeps me much drier. Weather on the coast is unpredictable. It can be sunny one minute and pouring rain the next (or even at the same time). Umbrellas are impractical for hikes or long walks—I will use one in the city but not in high winds. My hood kept my head dry yesterday on a walk, that's for sure!
My Thermoball hood is also non-removeable. I'd rather have the choice with that, but it's ok and turns out I used it yesterday, too, in a squall!
Some hoods fit me poorly. My neck is on the long side and my head is very long for my size. So sometimes they press my head down. I don't buy coats that make me feel that way. That rules out quite a lot of coats, in fact.
