Just to add some information/confirmation:

I started the DOCINDEX instance in minimal configuration mode using the following command:

"C:\Program Files\Microsoft SQL Server\MSSQL11.DOCINDEX\MSSQL\Binn\sqlserver.exe" -sDOCINDEX -f

Then I connected to the database and ran the following:

sp_configure 'show advanced options', 1

GO

RECONFIGURE WITH OVERRIDE

GO

sp_configure 'affinity mask', 0x00000000

GO

RECONFIGURE

GO

Which resulted in the following output:

Configuration option 'show advanced options' changed from 0 to 1. Run the RECONFIGURE statement to install.

Configuration option 'affinity mask' changed from 8192 to 0. Run the RECONFIGURE statement to install.

Then I stopped the server and restarted it from services.msc . It started and ran with no issues and the process and IO affinity were both set back to automatic.

I changed them back to Process on CPU 13 and IO on CPU 14. I went into services.msc, stopped the service, and upon restart it did the same "The service started then stopped" thing.

So it's DEFINITELY the affinity that's doing this, I just wish I knew why. At least now I have a way to get things working again without having to uninstall and reinstall the whole instance but it's still annoying.