A larger planet at same density means it has more mass. More mass, more gravity. More gravity, the spaceship needs to be faster to go around it, cancelling out the larger circumference.

Volume and circumference do not grow to the same power of the radius, i haven't checked if that still accurately cancels each other away or if the astronauts statement is an approximation.

Edit: Other poster linked mathematical proof https://en.wikipedia.org/wiki/Orbital_period#Effect_of_centr...

Solving for orbital period when at the surface, where d is the density of rock (googled):

d ~ 2500 kg/m^3

T^2 = (4 * pi^2 * r^3) / (G * M)

M = d * 4/3 * pi * r^3

substituting M into the equation for

T^2 = (3 * pi) /(d * G)