Contrary to the popular assumption, I don't think Walt shut his family out his Will...

During the reading of his Will, all we learn is that Walt left his house (which Mitch wants) to the Church and his treasured Gran Torino (which his granddaughter Ashley wants) to Tao. So most of us viewers just assume he left his family with nothing - But did he?
Walt had other possessions (tools, bank account, furniture, contents etc.) and the movie doesn't go into detail about what happens to them, but I think it's fair to assume at least some of those items were left to his two sons.

I know from our (and Walt's') perspective it might be very easy for us to hate Walt's' family and it's clear the relationship he has with them isn't good, but they're not evil, nobody is perfect, and Walt is certainly no saint himself. His standoffishness, prejudice, and lack of faith in anything is clearly what cultivated his sons to take a dislike and distance from him so much (and raise their own children differently), especially as fat-happy era boomers.
As the movie progresses Walt does improve and redeem himself in many ways - Not just learning to be more positive and tolerant, and bonding with the Hmong community, but also (in my view) realising his own imperfections and the fact that no one is truly perfect (see above).
It's a shame he never fully reconciled with his family, but I think by the end he had less animosity and more personal regret about not being the best father, as he stated in his eventual confession.