Cannot rebuild index, but there's no reason why not?
I skipped the code example, and jumped to what seems to be the real question here


  On top of that, a manual inspection of the index in question shows no text, ntext, image, xml or varchar(MAX), nvarchar(MAX) or varbinary(MAX). Could there be something I'm missing here?
  
  For the record, this is a clustered index.


You certainly are missing something, and the answer is in the body of the error message:  


  For a clustered index, the column could be any column of the table


For Online Index Operations there is a difference between a Clustered and a Non-Clustered index.
both may not contain a BLOB column in order to rebuild online,
but while a Non-Clustered Index "contains" the columns in it's definition (and included columns),
a Clustered Index "contains" the entire table with all of the columns.  

in your case there is a column named MEMO which is of data type text, ntext, image, varchar(max), nvarchar(max), varbinary(max), xml, or large CLR type.
this column is part of the table, and thus part of the Clustered index.  

Your options are:  


Perform the REBUILD operation offline
Upgrade to SQL Server 2012
Change the column's data type


Guidelines for Performing Online Index Operations