Let's say we have extracted lot of addresses from the wallet (say more than 2000), is there a quick and safe way to bulk check their balance?
You can use
my list of funded addresses locally (explanation in that topic) to see which addresses are funded. This can be done offline and in private.
Since you have an old wallet, you should note that funds sent to pubkey don't show up in a watch-only Electrum wallet.
You have "privacy" in your name. Dumping all addresses in Electrum or a block explorer kinda tells someone they all belong to one person.