When Mary gives this reason for not walking to Netherfield with Lizzy, I have to wonder what accompanying Lizzy to Netherfield would have meant.

Of course, it would have meant that Lizzy would not have arrived at Netherfield unaccompanied, one of the criticisms leveled against her by Caroline and Louisa. On the other hand, it would have been just as improper for Mary to have returned alone to Longbourn; so she would have been stuck with Lizzy and her patient at Netherfield until Lizzy returned home for dinner (at 3 p.m.?), presumably a boring prospect. Thus Mary would be sacrificing most of a day to accompany Lizzy. So I'm sure Mary had a practical reason (at least in her own mind) not to go.

In telling Lydia that she prefers reading to traveling, Mary is only being honest (and tactless). I do think Mary cares about her sisters, up to a point. She wants to fit in, so she plays airs at the piano when they wish to dance. She tries to share her wisdom with her sisters (poor unwitting Mary) for their edification and education. Mary is just socially inept and honest at the wrong times and in the wrong ways.

IMO "sisterly affection" is a goal at Longbourn. Lizzy's mention at Rosings of "promoting sisterly affection" may indeed be something she has heard at home; I can imagine her father saying it from time to time: it fits in with his allowing Lydia to visit Brighton to preserve peace in the house (not anticipating the effect on Kitty, apparently).