Any transactions are in public not private but all identities who use this address or who made any transaction are private.
So do you think if you send BTC to me or other forum members here did they know your identity or mine? Did you find it on the blockchain?
What makes you think about Chain analysis do to you? What they can only do is to track a transaction but they don't know who owns it and I don't think ChainAnalysis will track you unless you did money laundering.
Identities per se are not on the blockchain, but chainanalysis allows authorities to "triangulate" where a transaction came from, or where it went. It's not exact, but I'm pretty sure it works well enough to make it so that criminals no longer use Bitcoin, or if they do, they use a mixer.
And while
authorities won't go out of their way to track you down unless they have a reason to do so,
criminals and
marketers will use it to try to steal or sell something, or otherwise bother you

. In this context, using Bitcoin these days isn't very safe...