Could someone shed light on this? What's the rationale behind creating a stablecoin on a blockchain? Given that it's issued by a single entity with control over approved transactions, why not opt for a centralized database instead?[ ...]
The only advantage that comes to mind is the ability to transact at any time, since blockchain networks are available to access all the time.
My guess is that centralized stablecoins developed more or less to exploit the altcoin exchange market offering them an easy alternative to e-money (=fiat representations like PaypalUSD, for those who don't know). They were very easy to integrate into existing cryptocurrency exchanges, easier than traditional e-money because of their blockchain-based transparency, eliminating the risk of "chargeback fraud" (that's why those exchanges accepting PayPal usually charge high fees), and also easier than offering an on- and offramp via bank transfers. In addition, it allowed thousands of altcoin exchanges to offer a fiat-like product instantly, without having to maintain a huge fiat reserve, which would need other kinds of infrastructure.
So basically the main advantages are for exchanges, not so much for ordinary users.
Apart from what you already mentioned (which is also delivered by PayPal and similar e-wallet services), there are a few more advantages of stablecoins which also can appeal to some groups of ordinary users:
1) to store a fiat-like product with your own keys, not on an exchange;
2) to move a fiat-like product from one exchange to another one, for exampĺe for arbitrage purposes;
3) to integrate a fiat-like product into smart contracts, perhaps even into atomic swaps and Lightning;
4) to be able to store a fiat-like product without KYC in some cases.
Of course, the "not your keys not your stablecoins" principle is not the same as "not your keys not your bitcoin". First, stablecoin issuers can go bankrupt, and second, they can block your stablecoins even if you have a valid private key in most cases. So most stablecoins don't offer censorship protection.