The colorpicking in Illustrator has a wastly different function than colorpicking in Photoshop. It ultimately comes down to the differences in what your doing. In illustrator your basic primitive is a drawing object, in photoshop its a pixel. If you develop pixels its natural to be able to pick up pixel properties which are naturally limited to one color and maybe alpha. If you develop objects its not so clear.
You could take the stance that you in fact want to pick object properties, this is what both Illustrator and Indesign assume. But in reality youd want to be able to pick both object color, visible pixel color and anyhing in between. But short reason why you can not do this is that Illustrator does not really have a color picker at all it has a appearance picker but sice this is the closest equivalence its called the same. But since its not exactly a conguruent idea its s not behaving simillarily in all cases. For example you can use Illustrators color picker to pick a gradient since a objects color might be a gradient or a texture. Therefore it cant be used as such to pick just a color.
This is to say that theres no reason why the gradient context couldnt pick colors but then the dialog is not modal, and a new picker code could would need to be deviced just for this context. So if your willing to sacrafice functionality for design fragmentation and upkeep cost fine... But adobe hasnt really wanted to do this.
Dont expect two different things tonbe the same. SO ie most things there is a upside and downside in most things.