This struck me, also, as a fundamental obstacle. But Dyson was no slouch, so he must have considered this.

The calculation isn't hard: spherical surface 4(pi)R^2 times the thickness versus 4/3(pi)r^3 for the volume of the planet. So if the entire planet were used up to make the Dyson Sphere, its thickness would be:

1/3 r^3/R^2.

Since R is on the order of 10^8km and r roughly 10^3, thickness is roughly 10^9/10^16=10^(-7) km. or .1 millimeter. That's comparable to a sheet of aluminum foil or, maybe, a hot air balloon's sheath. Maybe the solar wind would keep it inflated once its complete. But there's the rub, how on Earth would you stablize it while deploying it?