There are no requirements in D1.1 for the person who writes or qualifies a WPS.  The "Contractor's Inspector", be it a CWI or other qualification, has the following responsibilities regarding the WPS:

6.3.1 Prequalified WPS. The Contractor’s Inspector
shall make certain that all prequalified WPSs to be used
for the work conform with the requirements Section 3,
Section 5, and the contract documents.

6.3.2 WPSs Qualified by Test. The Contractor’s Inspector
shall make certain that all WPSs qualified by test conform
with the requirements of Sections 4 and 5, and
contract documents.

6.3.3 WPSs in Production. The Contractor’s Inspector
shall make certain that all welding operations are performed
in conformance with WPSs that meet the requirements
of this code and the contract documents.

I write WPS' all the time.  Sometimes I even weld up my own PQRs, and boy does that make the welders a little uneasy.  As long as you have a good understanding of the code you're writing to and the fundamentals of required tests you should be fine; even if you are writing a WPS that meets all the requirements for being prequalified, you will gain a wealth of knowledge from performing a PQR.  I have seen folks try to write from just the code books, but I find a copy of B2.1 invaluable when doing any writing; PQR, WPS or WPQTR.

Now, occasionally a customer will have to approve use of a particular procedure.  That's when I go to my PQRs and show them everything, or if it's a new procedure then I'll run it for them.

As far as a CWI having to inspect the weld, that's not necessarily true.  I have no CWI at my shop, although we occasionally contract one to validate a procedure or test.