Blockchains aren't primarily financial tools. They are a political technology, and their natural field of application is the developing world.
The main problem a blockchain is meant to solve is lack of a trusted third party, which is at its root a problem of institutions, that is, politics. Bitcoin isn't used because it's convenient or scalable, but because it works as a rudimentary global financial system without having to trust any person or organization (at least that's the theory; poorly regulated financial intermediaries, like life, always find a way). The fact is that we do have a global financial system that it's relatively trusted, but bitcoin users — speculators aside — think the system checks don't work, think they work and want to avoid them, or some combination of both. I'm not judging.
Yet beyond those (huge) nooks and crannies in the developed world, there are billions of people who just don't have access to financial systems they can trust, and beyond finance, there are billions of people who don't have access to any kind of governance system they can trust. Honest cops, relatively functional bureaucracies, public records that don't change overnight: building a state that has and deserves a certain amount of trust takes generations, is always a work in progress, and is very difficult to even begin. Low trust environments are self-perpetuating, simply because individual incentives, risks, and choices become structurally skewed in that way.
Can blockchains solve this? No, obviously not.
But they can provide one small bit of extra buttressing, through a globally visible and verified public document ledger. Don't think in terms of financial transactions, but of more general documents: ownership transfer records, government contracts, some judicial and fiscal records, etc. Boring, old-fashioned, absolutely essential bits of information that everybody in a developed country just assumes without thinking are present, accessible, and reliable, but people elsewhere know can be anything but.
Blockchains working as a sort of global notary, set up by international development organizations but basing their reliability on the processing power donated by a multitude of CPU-rich but often money- and time-poor activists, would give citizens, businesses, and governments a way to fight some forms of mutual abuse. It won't, and cannot, prevent it, but it can at least raise the reputational cost of hiding, changing, or destroying documents that are utterly uninteresting to the likes of WikiLeaks, but that for a family can mean the difference between keeping or losing their home.
Even countries that have improved much in this area can strengthen their international reputations, and therefore their attractiveness for investments and migration, by this kind of globally verifiable transparency.
It's not sexy, it'll never make money, and it doesn't fully, or even mostly, solve the problem. It doesn't disrupt the business model of corruption and structural incompetence, and, best case, it'll put a small pebble in one or two undeservedly expensive shoes. Hopefully. Maybe.
But good governance is the core platform of a prosperous and healthy society. Getting it right is one of the hardest things, but also one of the most important we can try to help each other do.