Most investors don't hold their own keys anyhow, so this won't make a difference for most people.
But it's still important to have the ability to do so even if many don't feel a need to. If somehow that gets taken away and all that's left is exchanging on centralised platforms, Bitcoin will lose any purpose.
And a public ledger is not very... private. My GMail account is more private than Bitcoin because at least there would need to be a valid court order to breach my private email messages. Bitcoin transactions can be triangulated by anybody on the Internet.
Bitcoin is pseudonymous (don't be confused with pseudo-anonymous), meaning your public keys/addresses are your pseudonym, and it's up to you to take precautions of whether to connect your real identity with the pseudonym or not. So it can be private but doesn't have to.
As for Google account - they can be accessed by Google themselves if they choose to, so not much privacy there.