TL;DR: yes it can look more random than the plaintext message itself, but it won't help you against crypt-analysis and in the worst case it may introduce vulnerabilities.
I guess the most boring but most effective way is simply to look for meta data within the compressed message. E.g. the precise compression mechanism and compression level may well be within the decrypted message.
Generally you assume that the protocol is known (Kerckhoff's principle). Note that finding out the compression used is usually not hard, and you only have to do so for a single message.
Once you know the compression method checking the correctness of decryption is as easy if not easier; you can always first decompress before checking the correctness of the decrypted message.
Decompressing a message may well be faster than decrypting it. Performing the compression is usually slower as it needs to look for patterns. Compression will only add a single additional step to the crypt-analyst; it's not something you do to make your protocol more secure against brute force attacks. Basically you could compare it to adding a single bit to the security level at most.
An attacker may always use compression on a (set of possible) known plaintext and compare against that, resulting in a meet-in-the middle attack.
Compression in cryptography is a very dangerous thing. Some plaintext compress better than others and smaller plaintext generally lead to smaller ciphertext. So even if the plaintext is the same size, compression may confer details about the plaintext to an attacker. Take for instance a look at the CRIME attack against TLS for an example of this.
Compressed or inflated data usually does not look like random data, and it is a very dangerous to see it as a security feature. It is much more likely to add insecurity than security when trying to encrypt messages.
Important note: this answer is only valid for modern ciphers where frequency analysis of the ciphertext is not an attack vector.