PostgreSQL use NEW in query for INSTEAD OF trigger
NEW is a record, not a table. Basics:


Use NEW in FROM clause in Postgres trigger?


Slightly modified setup

CREATE TABLE product (
  product_id serial PRIMARY KEY,
  product_name text UNIQUE NOT NULL  -- must be UNIQUE
);

CREATE TABLE purchase (
  purchase_id serial PRIMARY KEY,
  product_id  int REFERENCES product,
  when_bought date
);

CREATE VIEW purchaseview AS
SELECT pu.purchase_id, pr.product_name, pu.when_bought
FROM   purchase     pu
LEFT   JOIN product pr USING (product_id);

INSERT INTO product(product_name) VALUES ('foo');


product_name has to be UNIQUE, or the lookup on this column could find multiple rows, which would lead to all kinds of confusion.

1. Simple solution

For your simple example, only looking up the single column product_id, a lowly correlated subquery is simplest and fastest:

CREATE OR REPLACE FUNCTION insert_purchaseview_func()
  RETURNS trigger AS
$func$
BEGIN
   INSERT INTO purchase(product_id, when_bought)
   SELECT (SELECT product_id FROM product WHERE product_name = NEW.product_name), NEW.when_bought
   RETURNING purchase_id
   INTO   NEW.purchase_id;  -- generated serial ID for RETURNING - if needed

   RETURN NEW;
END
$func$  LANGUAGE plpgsql;

CREATE TRIGGER insert_productview_trig
INSTEAD OF INSERT ON purchaseview
FOR EACH ROW EXECUTE PROCEDURE insert_purchaseview_func();


No additional variables. No CTE (would only add cost and noise). Columns from NEW are spelled out once only (your point 1).

The appended RETURNING purchase_id INTO NEW.purchase_id takes care of your point 2: Now, the returned row includes the newly generated purchase_id.

If the product is not found (NEW.product_name does not exist in table product), the purchase is still inserted and product_id is NULL. This may or may not be desirable. 

2.

To skip the row instead (and possibly raise a WARNING / EXCEPTION):

CREATE OR REPLACE FUNCTION insert_purchaseview_func()
  RETURNS trigger AS
$func$
BEGIN
   INSERT INTO purchase AS pu
            (product_id,     when_bought)
   SELECT pr.product_id, NEW.when_bought
   FROM   product pr
   WHERE  pr.product_name = NEW.product_name
   RETURNING pu.purchase_id
   INTO   NEW.purchase_id;  -- generated serial ID for RETURNING - if needed

   IF NOT FOUND THEN  -- insert was canceled for missing product
      RAISE WARNING 'product_name % not found! Skipping INSERT.', quote_literal(NEW.product_name);
   END IF;

   RETURN NEW;
END
$func$  LANGUAGE plpgsql;


This piggybacks NEW columns to SELECT .. FROM product. If the product is found, everything proceeds normally. If not, no row is returned from the SELECT and no INSERT happens. The special PL/pgSQL variable FOUND is only true if the last SQL query processed at least one row.

Could be EXCEPTION instead of WARNING to raise an error and roll back the transaction. But I'd rather declare purchase.product_id NOT NULL and insert unconditionally (query 1 or similar), to the same effect: raises an exception if product_id is NULL. Simpler, cheaper.

3. For multiple lookups

CREATE OR REPLACE FUNCTION insert_purchaseview_func()
  RETURNS trigger AS
$func$
BEGIN
   INSERT INTO purchase AS pu
            (product_id,   when_bought)     -- more columns?
   SELECT pr.product_id, i.when_bought      -- more columns?
   FROM  (SELECT NEW.*) i                   -- see below
   LEFT   JOIN product  pr USING (product_name)
-- LEFT   JOIN tbl2     t2 USING (t2_name)  -- more lookups?
   RETURNING pu.purchase_id                 -- more columns?
   INTO   NEW.purchase_id;                  -- more columns?

   RETURN NEW;
END
$func$  LANGUAGE plpgsql;


The LEFT JOINs make the INSERT unconditional again. Use JOIN instead to skip if one is not found.

FROM  (SELECT NEW.*) i transforms the record NEW into a derived table with a single row, which can be used like any table in the FROM clause - what you were looking for, initially.

db&lt;>fiddle here