I spent a lot of time looking for an answer to this question, but the majority of answers never explicitly explain why the client's public key is never used; it's always the server's public key that's used in SSL encryption. Similarly-phrased questions (such as this, this, and this) don't actually explain why the server's public key is not used. Rather, they just explain why public-key encryption is not used for the actual messages, or they explain some other topic instead.
So, my question is: why do we not use the client's public key in SSL encryption instead of using the server's public key? Theoretically, it shouldn't matter whether the client's public key or the server's public key is used, right, since in either case a master secret key is always established?