(Yes, I'm answering my own question, to record my solution)
I should also note, in addition to above, I couldn't use the SQL Native Client driver either.
In the registry, under HKLM\SOFTWARE\ODBC the \ODBC.INI branch contains any defined connections, and the \ODBCINST.INI contains records for the installed drivers.
I checked a similar development machine, and my ODBCINST.INI was missing A LOT of entries. I blame the ccleaner application that was recently used to clean up my system of junk.
After exporting the registry branch from the other computer, and importing over my existing keys, everything worked again.
Below are some of the core records, to generate a .reg script. You should get the full list from a similar machine to yours.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI]
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\Microsoft Access Driver (*.mdb)]
"UsageCount"=dword:00000002
"Driver"="C:\\WINDOWS\\system32\\odbcjt32.dll"
"Setup"="C:\\WINDOWS\\system32\\odbcjt32.dll"
"APILevel"="1"
"ConnectFunctions"="YYN"
"DriverODBCVer"="02.50"
"FileUsage"="2"
"FileExtns"="*.mdb"
"SQLLevel"="0"
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\Microsoft Access Driver (*.mdb, *.accdb)]
"UsageCount"=dword:00000003
"Driver"="C:\\PROGRA~1\\COMMON~1\\MICROS~1\\OFFICE12\\ACEODBC.DLL"
"Setup"="C:\\PROGRA~1\\COMMON~1\\MICROS~1\\OFFICE12\\ACEODBC.DLL"
"APILevel"="1"
"ConnectFunctions"="YYN"
"DriverODBCVer"="02.50"
"FileUsage"="2"
"FileExtns"="*.mdb,*.accdb"
"SQLLevel"="0"
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\Microsoft Excel Driver (*.xls)]
"UsageCount"=dword:00000002
"Driver"="C:\\WINDOWS\\system32\\odbcjt32.dll"
"Setup"="C:\\WINDOWS\\system32\\odexl32.dll"
"APILevel"="1"
"ConnectFunctions"="YYN"
"DriverODBCVer"="02.50"
"FileUsage"="1"
"FileExtns"="*.xls"
"SQLLevel"="0"
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\Microsoft Excel Driver (*.xls, *.xlsx, *.xlsm, *.xlsb)]
"UsageCount"=dword:00000003
"Driver"="C:\\PROGRA~1\\COMMON~1\\MICROS~1\\OFFICE12\\ACEODBC.DLL"
"Setup"="C:\\PROGRA~1\\COMMON~1\\MICROS~1\\OFFICE12\\ACEODEXL.DLL"
"APILevel"="1"
"ConnectFunctions"="YYN"
"DriverODBCVer"="02.50"
"FileUsage"="2"
"FileExtns"="*.xls,*.xlsx, *.xlsb"
"SQLLevel"="0"
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\SQL Native Client]
"UsageCount"=dword:00000001
"Driver"="c:\\WINDOWS\\system32\\sqlncli.dll"
"Setup"="c:\\WINDOWS\\system32\\sqlncli.dll"
"APILevel"="2"
"ConnectFunctions"="YYY"
"CPTimeout"="60"
"DriverODBCVer"="09.00"
"FileUsage"="0"
"SQLLevel"="1"
[HKEY_LOCAL_MACHINE\SOFTWARE\ODBC\ODBCINST.INI\SQL Server]
"UsageCount"=dword:00000002
"Driver"="C:\\WINDOWS\\system32\\SQLSRV32.dll"
"Setup"="C:\\WINDOWS\\system32\\sqlsrv32.dll"
"SQLLevel"="1"
"FileUsage"="0"
"DriverODBCVer"="03.50"
"ConnectFunctions"="YYY"
"APILevel"="2"
"CPTimeout"="60"