Yes well you can create blend shapes in 2 ways, by deleiting the originals and by not deleting the originals. The menutiem works on selection ONLY and ONLY if you chose not to delete the targets method. In wich case you choose target to remove and object to blend and hit the button. The other method is for when your really sure your done.
However if you deleted the targets it gets a bit tricky, since obviously you can not select whats no longer there! So you need to recreate it. This si how you do it:
duplicate the object that has the blend. connect it to your blend target slot that ewuates to your shape you need deleted. in this case connect the copy shapes .worldSpace[0] attribute to
blendShape1.inputTarget[4].inputTargetGroup[0].inputTargetItem[6000].inputGeomTarget; (this assumes what you show is what you have)
You can now go ahead and select the copy then blendshape and then the remove blendshape menuitem
This is ofcourse much easier to do in mel. Theres number of tools in the script repository that can help you bring the shapes back automatically so you dont have to bother with the connection.
in any case the mel to do this is:
proc deleteBlendshapeByIndex(int $idx,string $blend){
$item=ls -sl -tl 1;
$dup=duplicate $item ;
blendShape -e -t $item[0] $idx $dup[0] 1.0 $blend;
blendShape -e -rm -t $item[0] $idx $dup[0] 1.0 $blend;
delete $dup;
}
//in your case the index appears to be 4
//make sure you select the object and only
// the object that blends before you run this
deleteBlendshapeByIndex(4, "blendShape1")
...what am I doing wrong?
nothing really. Theres just no gui for this. (and please note your not really supposed to use the blendshape editor either nowdays you should have a better rig than that)
It never tells you what and when to select.
Well in your case there's probably nothing to select so telling you you cant do it wont help.
But they really really do ableit only once (yes it is assumed you have read the ENTIRE MANUAL). The idea is as follows you selet what you want to affect. This fact is firther bashed into your head lmel as the one to act on comes allaways last. So logically the order is same as when you made the blend shape. But here it helps nothing as no menu item can do what you want.
The Maya help is as hopeless as ever.
Well actually mayas manual is pretty good as far as manuals go.* Most users are juts hopeless at reading ANY manuals 
The purpose of a manual is not to tell do this and that. Because if such intructiins were enough they would do it themselves. They just tell you how it works.
*all manuals are in fact like this. You should read more manuals and their explanations will start to make somewhat sense. I dont mean this is the perfect for manuals its just that they CAN NOT LITERARY BE BETTER
