It is commonly held that if floating ice melts, then it makes no difference to sea levels.

This is true only of free-floating ice.

When a sheet of ice is attached to  land, basic principles of mechanics show that the weight is partially supported by that land.  A sheet of ice attached to land is a semi-cantilever.
A cantilever is a beam supported at one end only.
A semi-cantilever is a lever supported at one end, but with additional support at one or more places.  This support prevents the beam from sagging.  A striking example is in the struts placed over or under the wings of a monoplane, as shown clearly here in a blueprint.

A special case of the semi-cantilever is the buoyant semi-cantilever.  Imagine a floating pier or dock held in place by a horizontal hinge mechanism at the land end.  This will only suit an area with small tidal range.  In a rising tide case, the pier, acting as a beam, is subjected to compression at the top, and tension at the bottom face.  In a falling tide, these forces are reversed.  The long term fatigue effects are minimal.  In a case where constant wave action is a design factor, repeated cyclic and semi-cyclic stresses must be considered in determining fatigue failure probability.

Fatigue in a buoyant semi-cantilever is evidence of transfer of wave energy into the structure.  Wave energy transfer is minimal into any similar  fully buoyant structure.  The proportions of weight supported by the  land attachment and the water can readily be determined for rigid structures.

Where the buoyant semi-cantilever is a sheet of ice, even minor wave-energy inputs may become significant.  The energy of the waves causes visible distorsions, compression ridges and fractures.  The energy input, will, in all cases, be converted ultimately into heat.

In the case where land-attached sheet ice is treated as a semi-buoyant cantilever, the calculation is more complex, but the formula is already known.  The mathematical treatment of an ice sample in cantilever mode is given  in IHCR symposium on ice problems 1978 pdf, by Frederking and Hausler.  The additional complexities, omitted here, are examined in that document.

In summary, when calculating the probability that an ice sheet might detach, or the amount of sea level rise to be expected from such detachment, disintegrates or melts, the semi-cantilever effect must not be neglected.  In itself, mere detachment of an ice sheet without significant melting would cause an instantaneous propagation of a measurable sea level rise in my submission.

Footnote: this blog was written with the NOAA 'systems thinking' approach to climatology in mind.