The main disadvantage of too much rode occurs in crowded anchorages. You can't be swinging into people. Or boats.
The other disadvantage is when a change in wind direction may put you too close to a lee shore.
The length of chain does two things besides the one you described, which in itself is a great benefit. The anchor is designed to dig in, but will do so much more effectively when the length of chain adds weight to the end of the rode with the effect that the anchor is being pulled more parallel to the bottom.
The second thing it does is, counter-intuitively, act as a shock absorber. When a gust of wind shoves your boat to leeward, it may straighten out the rode, causing the end of the anchor shank to be lifted. But the weight of the chain must be lifted first, and the additional force required to lift it will be removed from the force being applied to pushing the boat to leeward. The decrease in velocity when the rode comes up hard against the cleat on your bow will reduce the chances of the anchor becoming dislodged. This is actually a problem they give you in introductory calculus...not that I remember any of that. You solve for the weight required to straighten out the rode given a particular force of windage on the boat.
Thirdly and more obviously, rope will be chafed or severed by rocks or coral on the bottom, whereas chain will be much more resistant to wear.