If the key fob battery is dead it still has a physical key that is removed from the key fob to unlock the doors and can still interface with the car via NFC to allow operation of the vehicle once inside (not op but I have a car with that going on so I’m making the assumption).

As an aside, if this is a topic that interests you, I recommend Code by Charles Petzold and The Code Book by Simon Singh. They are both exceptional books and very approachable.

While the public and private key are separate, they are mathematically related.

Say two large prime numbers are generated. Their product is used as part of the public and private keys. When you multiply two prime numbers together it is extremely difficult to “guess” what those two prime numbers were from the product, but if you have the private key you know the missing part of the equation that makes it difficult to reverse so you can easily reverse the equation and decrypt the message.

An encrypted message requires the key pair to decrypt. So if I’m sending you a message I use your public key to encrypt, then you use your private key to decrypt. Only you have your private key. If I were going to sign something, rather than encrypt it, I would use my private key, then everyone could verify it was me by using my public key.

https://www.khanacademy.org/computing/computers-and-internet/xcae6f4a7ff015e7d:online-data-security/xcae6f4a7ff015e7d:data-encryption-techniques/a/public-key-encryption

The way they explained things is how it works, do you not understand how public/private key pairs are used in encrypted communications?

Does this picture help?

I’m not trying to be an ass if my post comes across that way, I’m just unsure of your level of knowledge so I don’t want to offend by providing basic level info if it’s not wanted.