How wide are the aprons (sides) of the desk? If you make the internal leg braces nearly as wide as the aprons, you can greatly stiffen the leg mounting. Simply flipping the internal cross braces you show to be on edge, rather than on face, you can then screw them to the legs and stiffen their mounting. The legs you show are plenty sufficient to support. My work bench in my shop is 12 feet long and has eight legs like yours. The top is far thicker than what you show, but an elephant could tap dance on mine and the legs won't wobble. The aprons on my bench are 5 1/2" wide.
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"Sawdust is Man Glitter"