I agree that the public, replicable nature of the system is attractive, and there's some benefit in reusing known technology. But there still needs to be a central authority to "bless" certain actors in the system, else anyone could sign off on anything.
A good approach might be for a government health authority to issue "coins" to the various trusted entities in a product's chain of supply, and these then used as proof of stake for transactions in a private blockchain run for that purpose by everyone involved. That would keep bad actors out and would be very hard to hack, while remaining quite public for any interested parties. Now that's a cool idea, however very labour intensive to discover the provenance of any one item; certainly consumers in supermarkets would rely on some kind of query service - and there's your single point of failure again. Still, it's a cool idea.
None of this however would have stopped the tampering that launched this whole kerfuffle. Blockchains are excellent at ensuring the perfect reliability of digital data, such as numbers of bitcoins or digital kittens held by whoever owns abc123. They have no such effectiveness on physical products. I mean look at this:
> "You cannot open the can of baby food without breaking the label," he says.
The melamine was put in before the can was even filled!
If the audit is hashed and the hashes placed on the/a blockchain, then the audit cannot be modified without it becoming obvious.
Not only is the blockchain tamper proof but it is also a public record. So with traditional audit trails even if there are multiple copies you could still have the government seize or modify all copies. With the blockchain any such attempt at a cover up just isn't possible - once you've chosen to publish data publically and in a tamper prof way that suppresses calls of 'fake news!'.
So in that respect the choice to use the the blockchain can be seen as a genuine attempt to prevent corruption rather than paying lip service.
There's a trend of open sourcing blockchain code and the smart contracts that run upon it, meaning anyone can go and write a client to generate the tags or display the data without relying on a company's proprietary API or private IP. Despite this the guarantees are still there for the consumer.
In addition to that, the whole system is described in code that anyone can audit and is not susceptible to any human error (if done properly).
Ultimately this means that a company can't render all the RFID tags useless by going bust.
You're right, someone could go and host an API for charitable causes, but why do that when smart contracts are open source by default and do not require dedicated servers doing just that one thing.