Like ice cubes in a glass of water, the displacement doesn't change the level when melting occurs. So the icebergs and ice cap shelves seem like they would cause no rise.
The glaciers on land create a different scenario, but it seems that they would have to all melt in a very short time in order for the water stored in them to run directly into the oceans instead of infiltrating into the soil and replenishing the aquifer.
Am I missing something?

As far as Ice goes the only ice that counts is the Antarctic and Greenland ice sheets.
But even these controversial.
The Antarctic and Greenland Ice Sheets will recede one place and extend in another
And since continental plates float on a sea of magma the weight of that Ice carried on that plate presses down on that plate and will cause other plates to rise.
It is pure hubris for these scientists to say that they know the cause of rising sea levels in one particular place. The complexity of the problem is too great