Without an immutable shared database standard, it's difficult for processes like that to cross company lines. And supply chain is a fancy phrase for processes that cross many company lines.
A supply chain involves 3rd parties who have nothing to gain from this. If the issue is the middle man between A and C, blockchain isn't solving any problem. The distrust is with B, having a blockchained backed paper trail doesn't provide anything a database gives you right now.
You're asking for a single source of truth between A B and C. And saying you cannot trust A B or C, so you want blockchain to solve it, which needs to be created by someone, so you introduce D and now you have a central governing body who dictates everything for A B and C. Or you are A and you say 'hey B and C, you need to use this system we created cos we don't trust you!'.
Blockchain solves nothing in the supply chain.