Suppressing the adrenals is a good thing if you are sympathetic dominant(flight or fight), but not if you're parasympathetic dominant (rest and recovery). They're opposite branches of the Autonomic Nervous System (ANS). So yes, zinc would relax you if you're a high energy, nervous type.
People who are sympathetic dominant tend to be high energy, motivated, anxious, extroverted, and handle stress better. They tend to have higher pulse rates, large pupils, dry mouth and eyes, urinate less, insomnia, decreased gag reflex, and attraction to downers like alcohol, marijuana, and tranquilizers.
Parasympathetics tend to have low energy, apathetic, depressed, introverted, and handle stress poorly. They tend to have lower pulse rates, small pupils, lots of saliva and teary eyes, somnolence, urinate more, increased gag reflex, and attraction to stimulants like caffeine, ephedrine, or cocaine.
One side is not better than the other. Either extreme is not healthy. A balance is what you're looking for. Being on the extreme of sympathetic dominance can cause aggression, paranoia, mood swings, irritability, and anxiety. Definitely not a good thing.
Sympathetic dominant individuals can usually handle more volume in training, while parasympathetics can handle very little.
Sympathetics also do better on higher carbohydrate, less fat diets with an emphasis on fruits. Parasympathetics tend to do better with less carbohydrates and more fat, with little emphasis on fruits.
Sympathetics feel better when they use the supplements zinc, magnesium, potassium, and vitamin A. While parasympathetics do better with Calcium, sodium, iron, molybdenum, vitamin B5 and E.
Hope that explains it better.