Arguably, the published index, with broader coverage, is more relevant to assessing inflation trends in the United States than would be the harmonized index. As long as relative prices can and do diverge across countries, the harmonized indexes need to contain as broad a range of items as is practical.
As monetary union proceeds, then, it would be to the advantage of monetary authorities in the Euro area to have a consistent measure of inflation defined over a broad basket of goods and services that is measured according to established statistical methods.