Turn off auto qualification of table names when creating a view Problem Definition
I am trying to modified a shared development database, for a dev team to use for application development/testing.
Most of the collective work (table names, views, etc) are stored in the public schema, but I have set up schemas for each user to use as scratch space. However, the real goal would be for a user to use the objects in their schema if they exist and then fall back on the other objects, like how a search_path is used today.
Example
This would probably be better described with an example.  Let's say there's a database for an automotive repair application that a team is working on that has some tables and a view:
public.automobiles
public.parts
public.inventory
public.mechanics
public.schedule
public.v_repairs  -- view that joins fields from all tables above

This is working great, but let's say a developer (e.g., Sally) wants to test a new feature out, with her own dataset to check for visual feedback, or threshold testing.  She creates a table in her own schema, sally.schedule.  Because the default search_path is something like &quot;$user&quot;,public, any simple queries she creates, would first check her schema before public.  For example:
SELECT * FROM schedule LEFT JOIN mechanics USING(mechanic_id);

This would use public.mechanics and sally.schedule, when sally is connected to the database.  That's exactly the intended use, but when saving a view, it fully qualifies the table name by inserting the schema.  So if the same query above was created as a view in the public schema it would look like:
SELECT * FROM public.schedule LEFT JOIN public.mechanics USING(mechanic_id);

The magic of search_path is negated.  When Sally connects to the database to  call the view (SELECT * FROM v_mechanics_schedule), it ignores the sally.schedule table she created, and only uses the public one.
Is there a way to not have Postgres store the schema name for table/view objects when saving a view?

Note: this is something new I was looking into, but something I've never really had the need for, since developers can typically clone applications, replicate databases, and work within their own sandbox environment. No need for some clever collaborative schema setup
No. There is no way. I think there is a slight misconception of how things work.
Unlike functions, which store the function body as string (late binding!), a view parses the query and does not store the original text at all. All identifiers are resolved according to the current search_path and only their internal OIDs are saved (early binding!). That's also why SELECT * always resolves to the list of columns at the point in time when the VIEW was created.
What you see in pgAdmin or other clients when looking at the definition of a VIEW is a re-engineered SQL string. public.schedule will always be public.schedule and sally.schedule will always be sally.schedule (unless you rename table or schema). The (schema-qualified) names that are displayed are chosen such that the identifier is unambiguous with regard to the current search path.
This also shows when you change the names of columns, tables or schemas. The view keeps just working, because it does not depend on these attributions. The display of the definition is adapted dynamically.
Possible solution
If you want late binding you could use a function instead of a view and default to the current search path (one could SET the search path for the scope of a function to avoid that effect precisely.)
So instead of:
CREATE VIEW v_mechanics_schedule AS
SELECT * FROM schedule s  -- use aliases for short qualified column names in display
LEFT     JOIN mechanics m USING (mechanic_id);

Without schema-qualification, table names are resolved according to the current search_path at creation time.
You could:
CREATE FUNCTION f_mechanics_schedule()
  RETURNS TABLE (...) AS  -- you have to spell out columns
$func$
SELECT * FROM schedule s  -- this is typically error prone
LEFT     JOIN mechanics m USING(mechanic_id);
$func$  LANGUAGE SQL

Now, table names are resolved according to the search_path at execution time. Note that the same does not apply to the return type ( RETURNS TABLE (...)), which is determined at creation time. So this trick only works for compatible tables - the query has to return the same list of columns, only data types and the number of columns matter, column names are ignored (only the names in the definition of the return type are visible outside the function).
That's also why SELECT * to return values in the function body is typically unreliable. If you change the definition of underlying tables, the function breaks: The defined return type remains the same, but SELECT * resolves to a different column list ...
OTOH, if you declare a function as RETURNS SETOF some_table (different syntax variant), a functional dependency is registered and the return type of the function is bound to the table definition, so SELECT * FROM some_table would make more sense.
This is also the reason why it is unsafe to have SECURITY DEFINER functions without fixing the search_path: Any user with the TEMP privilege can create a temp table that hides other tables because pg_temp comes first in the search path by default ...
Related:

Permission for sequence in another schema

You really need to understand the underlying mechanisms to play with this.