The benefit is lower tracking distortion (but nothing like as good as a linear tracker in this regard). However just being longer does not guarantee superior performance - lots of other design elements impact as least as greatly as length (note: arm effective mass is increased).
You don't need a bigger platter (records are the same size), but you do need to mount he arm 3 inches (approximately) further out from the record centre than a 9 inch arm. Many decks will not accommodate such arm positioning which is partly why 12 inch arms quite rare.
That's interesting about the platter, because most of the tables that use 12" arms now have larger platters. Nott has a table with an enlarged platter, and so does the SME 20/12.
I've listened to the Shindo table with a 12" Shindo-modified SME and it had a regular-sized platter.