He is just a top notch professional. Could have jetted to the tropics but probably thinks about his rehab routine and its better to stay back to get ready...the challenge of getting minutes in our minutes as everyone returns to fitness and that is not going to be achieved by flying out.
I am also sure he would have asked Klopp to play(if recovered enough) if Klopp had not drawn a line in the sand about no senior players participating.
In a previous post, I speculated that players who have school going kids might not be jetting out since they would have to take kids out of school. I think Milner is in this category and also might speak of the professionalism he is putting in his kids - "no school days off"
of course just speculating but that would be so Milner.