I've been asked by a few of our friends here to explain what metacentric height is and why it is important in estimating warship stability. OK, here goes. I've left the mathematics out as much as possible - if anyone is interested in the geometry that goes with this lot, let me know and I'll point you to it.
The forces acting on a ship floating in the water are primarily those of gravity (the total weight of the ship) and buoyancy (the weight of water displaced by the ship hull). These can be visualized as the ship's weight pushing it down in the water while the ship's buoyancy pushes it up. Under normal circumstances, these act along a vertical plane that passes through the center of the ship's hull.
The next step in working out the ship's metacentric height is to calculate the ship's center of gravity. This is determined by calculating the moment of all the various parts of the ship relevant to a reference point. This gives the center of gravity (G) which is the exact center of the ship in terms of the ship's weight distribution. Then, we calculate the ship's center of buoyancy (B) which is the exact center of gravity of the volume of water displaced by the ship's hull. (G) and (B) both fall on a line that passes through the exact center of the transverse section of the ship's hull and points vertically downwards. This line is called the Transverse Metacenter.
In most cases, the center of buoyancy is below the center of gravity. (G) and (B) are expressed in terms of height above the ship's keel. To calculate the metacentric height (B) is subtracted from (G) to give the metacentric height (GM). Ships have two metacentric heights, one being the transverse metacentric height (GMt) calculated using the transverse cross-section of the ship and the other the longitudinal metacentric height (GMl) using the longitudinal cross section of the ship. (GMt) controls the recovery of a ship from listing, (GMl) covers recovery of a ship from pitching.
Lets assume a ship has taken a hit and is flooding along one side. This causes her to list to an angle (s). The transverse metacenter through the ship's hull no longer coincides with the original line but is at an angle to it. Unless there is a dramatic change in the ship's shape at the waterline, the center of gravity (G) remains in place and the new and old transverse metacenters cross at this point. The center of buoyancy shifts to the side where flooding is taking place. This displaced center of buoyancy is designated (Z). The distance between the original center of buoyancy and the new center of buoyancy is defined as the transverse righting arm (GZt) and is calculated by (GZt) = (GMt) x sine(s). Equally, a ship's recovery from digging her bows in (for example in heavy weather or due to hitting icebergs) is defined as (GZl) = (GMl) x sine(s). This shows why the Iowa's were so wet forward; because they had very fine lines, the mass of water displaced by the bows was not great so digging the bows in did not displace the center of buoyancy much so little righting arm existed. On the other hand, this gave her the slow, easy motion in heavy weather that veterans of the ship have described.
This makes metacentric height a very useful index of ship's stability. Its used in this context is based on the assumption that adequate GM in conjunction with adequate freeboard will assume that sufficient righting moments exist at any practical angles of heel. However, freeboard is a limiting factor. If the ship's list reaches a point where the freeboard on the sinking side of the ship reaches zero (that is, when the water comes over the ship's side and starts to flood over the deck), all bets are off. The moment water starts to flood over the deck, (GZt) drops dramatically since the effective beam of the ship declines very quickly with each additional immersion increment. Thus, acceptable GM is directly related to freeboard, explaining why ships with limited freeboard (such as destroyers) also have much lower acceptable GMs than ships with great freeboards (such as battleships).
A large GM gives a large righting arm. This means a ship will snap back from a roll quickly but will also roll easily. This gives a violent motion (very much like a WW2 DE). In contrast a low GM gives a small righting arm which means the ship will roll slowly but return even more slowly.
With a DD, the problem is that too fast a roll will swamp the freeboard quickly. When the deck edge dips under and water starts to climb up the deck, the ship's effective hull shape changes very quickly and very radically. Inter-Alia, this changes the position of the center of gravity drastically, unpredictably and very, very, quickly. The result is usually the destroyer rolling over (see the three DDs lost in the Great Typhoon of 1944). Also, as you point out, in a shallow hull, there is simply less volume to accommodate a great GM but you wouldn't want to anyway. In short, roll period (to which GM is a pointer - but not the only one) has to be proportional to ship size.
By the way, GM is only an indicator. To determine the stability of a ship accurately, the designers have to do inclining tests, deliberately listing the ship to varying angles then doing strings of calculations. This often reveals some nasty surprises - Evil Men have spread rumors about such trials ending with the designers yelling "oops" as the test ship rolls over. I have never found such cases documented.
Which doesn't mean that they haven't happened.