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.