I'd be interested to know if this condition can be cured using cells from a baby's umbilical cord, which apparently is effective as a source of stem cells?
There are many possible sources of cells for clinical use, via methods often call regenerative medicine. You are correct that umbilical cord stem cells are one, as are embryonic stem cells, adult stem cells, induced pluripotent stem cells (iPS cells) and loads of cells that are 'differentiated' cells rather than stem cells.
There is no magic bullet, one size fits all and as we develop these techniques one type of cell may be best for one clinical application, with a different type best for a different application. Currently we don't know which works best as there is far, far more research to be done. Research is ongoing looking at all the different cell types in labs across the world.
Therefore to ignore one particular cell type is foolish as that might be the best type and we would therefore be cutting off the opportunity to develop the best treatment.
There is another big reason why researchers use embryonic stem cells - specifically because they are the cells which form the embryo and all the tissues in our body eventually. That reason is to study the earliest developmental processes, and in particular what goes wrong in many congenital diseases. That research cannot be carried out using adult stem cells or umbilical cord stem cells as their developmental stage is too late and for umbilical cord stem cells aren't actually part of the developing baby.
Another obvious problem with umbilical cord stem cells is more pragmatic - unless you have had them banked down at birth, there isn't a person-specific cell source available. And for the vast, vast majority of us we don't have our umbilical cord stem cells banked down and can, obviously, we can never remedy that.