Neither of which I suggested getting rid of above (though to be clear, proof of stake and proof of burn both exist). Transactions are especially handy when you move beyond merely recording the drivers' licenses themselves and actually start recording things like, say, traffic tickets or new endorsements or what have you - and that handiness is significantly improved further when agencies are sharing a single public blockchain instead of trying to maintain umpteen different internal databases explicitly tracking each other.

Even taking your assumption that a blockchain and a database mirror are "literally the same thing", you're still missing the key difference: that the former gives you the benefits of the latter (or at least the ones relevant to this context) with significantly less effort. It's similar in comparison to, say, using IPFS v. mirroring an FTP server; yes, you can create a bunch of FTP mirrors, and reinvent a poorly-specified ad-hoc substitute for half of IPFS, but it'd be a heck of a lot easier and more robust to... just use IPFS.