But if someone chooses a ciphertext (without knowing the key), how could they get the checksum right (which is also encrypted)? If the checksum was wrong, the system would refuse to decrypt the corrupted message.

No. There's no such thing as "refusing to decrypt a corrupted message"; cipher cores either produce an expected plaintext or, if the message is corrupted, an unexpected plaintext. Attackers can use changes in behavior based on the differences between different unexpected resulting plaintexts to infer the original plaintext.

My question was why is a proper MAC better than appending a checksum to the plain text and then encrypting. With an appended and encrypted checksum, the system could easily reject corrupted messages, that's the whole point of a checksum in the first place.

So why is a proper MAC better than appending a checksum or hash to the plaintext and then encrypting? Or maybe I am misunderstanding something?

I've answered that already. The MAC allows the cryptosystem to reject messages with corrupted ciphertexts. If you don't do that, it can be possible to use controlled corrupted ciphertexts to learn things about the plaintexts of messages, and not just via CBC padding oracles.

Doesn't encrypt then mac allow you to reject messages prior to decryption?

Yes. That's not what we're talking about here, though.

how would the plain texts differ?