Not quite. The analogy works for secret keys (what you're calling "symmetric keys") but a private key is unlike real world key analogies because as you'll quickly discover if you deploy real world keys in a high security environment people can examine the lock to figure out the correct key. The whole point of private keys is that this cannot work in a public key system. That step does not exist.

It is true that publishing your private key is bad but you'd hope the name makes that pretty clear. Despite the way I remember (U2's "The Fly" lyrics, "A secret is something that you tell one other person, so I'm telling you, child") people generally do not understand that the whole point of secrets is that at least two parties know, which means you might always be betrayed by somebody you think is keeping your secret. For a private key it's easy, don't tell anybody, nobody knows, you can't be betrayed, done.

For example Hacker News learns my password to this web site every single time I sign in because that's just a secret. We've known how to do better for decades but only a handful of systems I use (e.g. Google) do so and all of them have a "traditional" password option which is like discovering your aeroplane still has a smoking section in 2026.