Having a connecting arch across the top is both decorative and will greatly reduce the cantilevered forces trying to bend the hinge side upright. Bonus if one of the uprights has a diagonal brace to the ground (perhaps if it connects to a fence). A diagonal brace with an arch top will make for a strong frame structure.

However there will still be a bending load when the gate is open.

Something as simple as a top strap to fasten that upright at it's top to the wall will also create a sturdy structure.
I don't think bottom brackets on their own will hold up over time unless they are very sturdy.
You should attach the uprights to the ground either way.