Not sure about this, but it is reasonable to expect the hash of the seed to need to be more secure than the hash that generates the result. If anything, the HMAC-SHA512 for generating the result is overkill, compared to the SHA256 that is used to hash the server seed and is displayed to the user. The hash displayed to the user should considered to be more vulnerable and exposed to possible brute force, as you've mentioned.
@OP: Is there an actual technical reason for using the less secure SHA256 for the server seed, or would it be possible to standardize it to SHA512?
What are you not sure about? Bitcoin mining uses sha256, the asic hardware performs these hash calculations. You could just tell it to perform it on arbitrary inputs. But I agree, it is weird that the site uses sha512 for the case where comparatively less security is needed, and sha256 for the case where you really need to be sure of security (would expect the opposite, if nothing else).