Seagate is correct. Putting a hash on the website doesn't improve security at all because anyone who can change the download can also change the web page containing the hash. ... A company like Seagate doesn't rely on volunteers at universities to distribute their binaries so the technique is pointless.
There are many possible attacks. A hash on a website is not invulnerable to a rogue employee at Seagate (or one "just following orders").
A hash protects against a rouge insertion at the endpoint. Like if your PC is compromised by an attacker and then you pull the hard drive and [assuming there's a way to get a hash from SMART/ATAPI) you can compare the hash of the firmware that the drive is running to the list of published firmwares at the vendor's site. If the attackers are only modifying a small subset of drives, this works fine - they can't also intercept the check to the vendor's site - not unless they've broken TLS and/or have malware on every possible machine.
A tool to verify the firmware is poetically impossible to write. What code on the drive would provide the firmware in response to a tool query? Oh right ..... the firmware itself.
Well, today you can pull the image from JTAG, or so the experts have said (you can verify the firmware directly from memory with a hash if you have moderate funding). There's all sorts of talk about how ATAPI is write-only for firmware because the vendors don't want their competition to get their code and decompile it. This appears to be nonsense, as any other drive vendor already has the debug tools to pull such things from memory, and extracting it from an update isn't that hard - if a 16K DOS update utility can extract it, so can a multi-billion dollar R&D company.
To make it work you need an unflashable boot loader that acts as a root of trust and was designed to do this from the start. But such a thing is basically pointless unless you're trying to detect firmware reflashing malware and that's something that only cropped up as a threat very recently. So I doubt any hard disk has it.
They most certainly do not. So, here we are at today and need a way forward. There are a few ways forward, a fistful of crypto protocols to choose from to ensure future usefulness of hard drives for security applications, and INCITS/SATA-IO ought to be having emergency meetings _right now_ because this (NSA/GCHQ) is a major threat to the industry. The vendors may need to move operations outside of five-eyes to remain commercially viable.