The downside to such a system is that it either requires a central authority to manage the contracts, and have some sort of stick it can hit you with, or it can physically be changed. With a clay tablet contract for example I could destroy the tablet, or it could be held by the third Mesopotamian government and enforced by them to throw me in debtors prison or something if I refused to pay. This necessitates a central authority with something like "legitimate state monopoly over the means of violence" in a best case scenario to keep bad actors from violating their agreements. In a less than best case scenario, such a monopoly allows for exploitation of it's power.
A decentralized ledger allows for agreements to be made without the risk of the agreement being broken by the other person or the central authority. Precious metals fulfilled the same role previously, as an ounce of silver is an ounce of silver, regardless of what may happen in the future.
Frankly I have no idea if anything will come of it in favor of existing systems. Perhaps on the whole the existing system is more effect. But personally it's pretty exciting from a historical/economic perspective.