@JustA_ Since he's an old man, I don't think getting it on with her would work. =/ That and he might eventually die. The only saving grace would be that if he suffers from an incurable disease, that his Duke sends him off...
Re-reading this from the start, I see why Rob was sent with MC.
His mother actually does love MC, she just sucks really bad at showing it. She wrestles with issues of not being able to be warm and able to show emotions. However, Rob is clearly an excellent butler, even MC says that he is able to do all the work very quickly and efficiently. MC's mom sending Rob was probably her own way of trying to give him the best care, not so much tossing aside Rob because he's old.