As is the case on Earth, the Moon's crust is less dense than the material that comprises the Moon's mantle or core. On the Earth, this means that areas with thick crust rise above sea level. (These are our continents.) On the Moon, this means that the Moon's center of figure (it's center based on surface topography) is offset by a couple of kilometers from Moon's center of mass, with the center of mass closer to the Earth than the center of figure.
This center of mass to center of figure offset does not mean much with regard to the Moon's stability. What matters much more is the distribution of mass top to bottom vs side to side vs front to back. To be stable in a tidally locked orientation, a moon's principal axes deduced from the moment of inertia tensor should be very close to the set of axes as seen from the planet.
Moreover, a very specific ordering of the moments about the principal axes as deduced from the inertia tensor needs to exist. For the moon to be in a stable tidally locked configuration, the axis of rotation needs to be more or less orthogonal to the moon's orbital plane, the moment of inertia about the rotation axis needs to be the largest, and the moment of inertia about the line from the center of the planet to the center of the moon needs to be the smallest, leaving the moment of inertia about the side-to-side as the intermediate axis.
This most definitely is the case with the Earth's Moon. The angular difference between the Moon's geometric axes as seen from the Earth and the Moon's principal axes deduced from its inertia tensor (in technical terms, the angular difference between the Moon-Mean Earth frame and the Moon's principal axis frame) is a few hundredths of a degree.
That the Moon's far side has a much thicker crust than does the near side is a third order effect. The Moon would be nearly as stable as it is now if it was the Moon's far side with its much thicker crust that faced the Earth rather than facing empty space.