I am learning about crytography and the differences between the public and private keys encryption examples and I was wondering if it is possible to have a private key encryption scheme that is completely insecure? I have not been able to find what that could look like but I'm curious? Or is this not possible? Is the encryption scheme always secure with private keys?
Any encryption scheme can be insecure if the key space is small enough.
For example, you could perform encryption with an 8 bit RSA key. For a key of that size, it's trivial to determine the private key given the public key.
What if the key is completely secure and protected, would there be any way to have a completely secure key but insecure scheme?
Yes, certainly. For example, consider the following encryption scheme:
The key is a 256 bit (or, heck, 512 or 1024 bit if you want) string chosen uniformly at random by a cryptographically secure true random number generator, stored securely in a locked vault deep underground, with multiple armed guards watching the entrance 24/7.
The encryption method doesn't use the key for anything, and instead encrypts the data using rot13.
The decryption method is the same as the encryption method.