What repair options are available for Thinkpad Stack hard drives?

[…] is a third-party data recovery service appropriate for me?

Data recovery service does what it says: they recover your data, not fix your device.

One acceptable solution I can think of off the top of my head is to gut the hard drive, put a new one in, connect it to the pogo peg thingies, and reseal the case.

That sounds like a plan. It's most likely using a bog-standard SATA HDD that can easily be replaced. Confirm that that's the case before buying a new drive.

Getting a new drive to work may be a challenge if the device makes some assumptions about drive's format. Inspecting the original drive could help, but given that it's dead that's not an option. So I'd suggest to try it on an unused drive you may have lying around or a cheap used one before buying a proper replacement.

Then take this into consideration:

• Match the form factor. If the original HDD is 2.5", you won't fit in a 3.5" drive or vice versa.
• Match the rotational speed. It will most likely be either 5400 rpm or 7200 rpm. 5900 rpm is also not unheard of. HDDs with higher rpm will be faster, but they will draw more power and the device may not be designed to provide more than the original HDD is expected to use.
• The larger the drive capacity, the faster sequential transfers will be. That won't necessarily translate to random access though.
• Avoid SMR drives. These are cheaper, but their data storage method makes them problematic in some use cases.