Is there a term for cryptographic attacks where you can verify decryption without assuming any known structure on the plaintext?
You're missing that using a KEM for hybrid is not encrypting random garbage.
Say you have a chain of ciphertexts $C_1||C_2||\dots|| C_n$, where $C_1 = \mathsf{Enc}_K(k)$, where $k$ is an AES key, and $C_2||\dots||C_n$ is the result of encrypting some stream of plaintexts under the AES key $k$ in some mode of operation.

If we can compute $k = \mathsf{Dec}_K(C_1)$, we can easily check if we decrypted correctly by decrypting $C_2||\dots||C_n$ under the aforementioned mode of operation.
Assuming this data has any structure (or any way to distinguish it from random garbage essentially), one can bootstrap that to a method to distinguish $\mathsf{Dec}_K(C_1)$ from random garbage.

If someone did encrypt (using the block cipher mode of operation) some sequence of random data $P_3||\dots||P_n$ (one block is used for the IV in the mode of operation), we could no longer use the aforementioned method to distinguish a correct AES key.
Then the linked comments would still be problematic (from the perspective of an attacker) --- there's no way to distinguish a bunch of random garbage from random (provided its distributed the same), so its essentially hopeless to ask.

This is essentially making the point that just because $k$ is (hopefully) uniformly random on some domain by itself, the rest of the ciphertext blocks are highly dependent on it, which can be utilized to develop a "correctness test" for guessing $k$.
So $C_1$ satisfies the "random garbage" property, but $C_1||\dots||C_n$ (jointly) doesn't.

Your second question is interesting, but I can only comment on the first question unfortunately.