Sounds a like silly question but I played a hand that makes me think about it.

Hand : - vilain open mp and 3 barrel on Ac Kd 7c Tc Ah

          - hero : flat sb, call flop+turn and check river with 7d7h

Could I shove for value ? vs this bet-calling range river [AA,KK,AK,AT,(KQ,KJ,QJ,89,76,65):cc], 77 has 24% so I can't value bet. If I have AT:xx and some AK in my own range it's ok I can use it to XRAI for value. But imagine I haven't those combos in my range : 77 is top of my range but I can't raise it for value .... Sounds bad for me both in practice and theory  ....

What do you think about these situation ?