The whole point of bitcoin is that anyone can verify it, it's supposed to be trustless. If transactions were private, how could you trust anyone claiming to have BTC? How could you trust that BTC being sent to you in fact exists, instead of the sender just making up a transaction spending outputs which never existed? That's why all full nodes verify all transactions since the genesis block, to ensure all bitcoin being sent are legitimate.

A separate blockchain is possible but perhaps not in the way you think. It's possible to use sidechains or second layer networks like Lightning, but to make a separate blockchain itself would be to create a new currency completely, it would be an altcoin not bitcoin