What am I understanding wrongly?
You are mixing up several distinct uses of cryptography, several different purposes:
- proof of integrity
Cryptography in General
If you want to send a private message to someone, in a way that no-one intercepting the message can read it, you typically encrypt the message using a symmetric (secret-key) algorithm and a newly created secret-key. You then encrypt the secret-key using the intended recipient's public key. This ensures that only the intended recipient can decrypt the secret-key for the message by using their private key (which no one else knows). After decrypting the secret key, the intended recipient is able to decrypt the message using the symmetric algorithm.
Note that you only use a symmetric algorithm to encrypt the message because such algorithms are far far faster than asymmetric (public-key) algorithms. It is primarily for efficiency, not for any other reason.
If you want to send a message to someone and provide proof that you sent the message, you use a different procedure. You create a cryptographic hash of the message using a hashing algorithm. You then encrypt that hash with your private key (that only you know). The encrypted hash is sent with the message. This encrypted hash is known as a digital-signature. The recipient of the message can use your public key to extract the hash and compare that with a freshly computed hash of the message, if the hashes match it is proof that you sent the message.
Types of Encryption
Asymmertic means you use different keys for encrypting and decrypting. There is both a private key and a public key. hence the name "public-key cryptography".
By contrast, an example of a symmetric (not asymmetric) algorithm is DES which is a symmetric encryption algorithm because there is only one key, the same key is used for both encrypting and decrypting. The key is a shared "secret-key" - hence the names "secret-key cryptography" or "shared-key cryptography".
Cryptography in Bitcoin
From what I've read, Bitcoin uses
- Hash algorithms
- Digital signatures
- ECDSA for signatures on transactions
Note that ECDSA is asymmetric
Note also that Bitcoin cares about anonymity and not about privacy (the blockchain is public not private). Therefore Bitcoin does not use encryption for privacy purposes. Your Wallet software might use encryption to hide your secret keys behind a password - but that is not part of the BitCoin spec.