It's all about trade offs, and conditions. This might work for short rides with light winds. But I can see it becoming impractical for longer rides, or if it's windy or even just a bit warm.
Even in the most miserable, coldest November and winter rains, I've given up and adapted to wet legs. I do all I can to keep my torso warm, because that seems to have the biggest effect on me. I try to keep arms dry if it's really cold, but otherwise I don't worry about them.
As for the legs, I wear layers that offer some insulation even when wet, and find that they produce enough heat that I can live with them being cooled.
In fact, I've found that the key isn't to be dry, but to separate the warm water inside from the cold water outside.
Lastly keep in mind, that trench coats aren't designed to resist driving rain. When you ride, the rain blasts against your body driven by your speed (you hit the drops, instead of them hitting you), and I,m not sure a trench coat won't soak through pretty quickly, having ride not only wet, but weighed by a bulky sodden coat.
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FB
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WARNING, I'm from New York. Thin skinned people should maintain safe distance.