Question about schema-level permissions

I've created a schema called Indexer and a user called Indexer.  I
then grant Indexer ALL on said schema:

GRANT ALL ON SCHEMA Indexer TO "Indexer";

Next, I attempt to INSERT into Indexer.ParseErrors, I get a permission
denied error message.  However, if I specifically grant Indexer INSERT
permissions:

GRANT INSERT ON Indexer.ParseErrors TO "Indexer";

Then everything works.

Am I missing something?  Doesn't GRANT ALL mean that user can do
anything they want with objects in that schema, including inserts?
Thanks!

Mike