Nothing in particular, other clients can already check every script types that Blockchain(dot)com used within its lifetime.

The difference is just you'll be relying on your own copy of the Bitcoin blockchain if you used Bitcoin Core to rescan.
And even with you own copy of the blockchain, you still need the correct public key(s) to find which is which.

Note: "Blockchain" online wallet and the Bitcoin "blockchain" are different things.
I normally use capital letter on the former to identify that it's the company that offers an online wallet/exchange service.